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Chapter 5 Some Methods
C H A P ~5R
SOME METHODS
In this chapter we deal with general methods for evaluating or estimating, from below, the critical determinant of a given set in k". We give various examples and, in a number of cases, also determine the critical lattices. However, the systematic treatment of some important classes of star bodies will be postponed to the next chapter. In general, we can say that the critical determinant can be determined effectively only for a limited class of bodies; as to star bodies of dimension n > 3, as a rule, we can only derive more or less sharp estimates. The last two sections partly concern inhomogeneous minima.
29. The critical determinant of a two-dimensional star body. Methods of Mahler and Mordell 29.1. We know already a method by means of which we can find the critical determinant and the critical lattices of a bounded convex star body K in the plane; in fact, the lemmas 22.1 and 22.2 tell us that the critical lattices are determined by the inscribed affinely regular hexagons of minimum area. We now deal with a more general method which applies to arbitrary bounded star bodies in R2.This method, too, was developed by Mahler [29a, 29bl. It is mainly of theoretical interest. Let S be a given bounded star body in R2,with boundary R, and let A be a given lattice having two independent primitive points xl, x2 on R. The points xl, x 2 need not form a basis of A . But there is.a point y 1 such that {xl,yl} is a basis of A. Take such a point y l . Then { X I , y} is a basis of A if and only if y = + y l kxl, for some integer k and some choice of the sign. Moreover, if {xl, y } is a basis of A, then the point x 2 can be expressed as follows:
+
x2 = u l x 1 + u 2 y (u l, u2 integers with u2 # 0). (1) By choosing y suitably we can obtain that (2)
u2
> 0, 0
Ul
< u2.
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THE CRITICAL DETERMINANT OF A TWO-DIMENSIONAL STAR BODY
331
Clearly, y is determined uniquely by the requirement that {XI, y } is a basis of A and that the coefficients ul, u2 in (1) satisfy (2). It is not difficult to express the points of A in terms of XI,x2. Indeed, if y is determined as above, then the points of A are just given by the points x = ~ ~ x ' + t=~U ~; ~y ( ( ~ ~ U ~ - Z ~ +~ ZU ' ~~ )XX~ ~)
( Z ) ~ , I I integers). ~
Thus A is the set of all points x = u2- 1 ( W 1 X ' + W 2 X 2 ) ,
(3)
where w l , w2 are integers with w1 + w2 u1 = 0 (mod u2). We now state and p rwe a theorem which says that, for given x', x2 E R, there are only finitely many S-admissible lattices containing xl, x2 and that these lattices can be determined in a finite number of steps. The admissibility of A entails that XI,x2 are primitive points of A. In the following, fGr p > 0, we denote by C , the disc 1x1 5 p .
Theorem 1. Suppose that C, c S c C,,, where 0 < p < u, and let xl, x2 be two independent points on the boundary R of S. Then the S-admissible lattices containiflg XI,x2 are found among the lattices A having a basis {XI, y } such that the coeficients u l , u2 in the representation (1) satisfv (4)
On the other hand, such a lattice A is admissible, provided int S does not contain a point x # o of A such that (5)
lWil
6 a2{d(A)}-'
(i
=
1,2),
where w l , w2 are determined by (3). Proof. First let A be an S-admissible lattice containing x', xz and let y , u l , uZ be determined such that {XI, y } is a basis of A and ul, u2 satisfy (1) and (2). Then ldet (xl, xz)I = u,d(A) 2 u2d(C,) = ,/+
- u2p2.
We also have lxil 5 u (i = 1,2). Hence, u2 5 ,/$* ( ~ / p ) Thus ~ . (4) holds. Now suppose, in addition, that A does not contain a point x # o
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of int S such that Iwi( a'{d(A)}-l ( i = 1,2), where w l ,w2 are determined by (3). If x is a point of A such that IwlJ > a2{d(A)}-', then ldet (x, >I)'.
= u l ' . Iwl det (x*,I>)' .
= Iw,ld(A)
> a' 2 alx21.
and so 1x1 > a, so that x $ S . Likewise, x $ S if x e A and lw21 > > a2{d(A)}-'. It follows that A is admissible for S. This proves the theorem. Mahler considers in more detail the case that R consists of finitely many analytic arcs R 1 , . . ., R, separated by points b', . . ., bP on R. Thus, for i = 1, . . ., p , a variable point x E R i is an analytic vector function of a real parameter a. This parameter a can be chosen in such a way that the derivative x' = x(a)' does not vanish and so determines the tangent to R at the point x, for x E R\Z. Here, Z is the set {b', . . ., be}. Now let A be an extremal lattice for S. It contains two independent points x ' , x2 of R ; these points determine, on their turn, two integers u l , uz . The integers u l , u2 are bounded, by theorem 1. Mahler now proves that there is a finite set of pairs of equations with the following properties: 1". each pair xl, x 2 associated with an extremal lattice for S, with fixed values of u l , u 2 , satisfies one of these pairs of equations; 2". the pairs of equations can be written down explicitly; 3". each pair of equations is satisfied by at most finitely many pairs x', x2 and at most finitely many continuous sets of pairs xl, x 2 on each of which det ( x ' , x2) is constant. To give an idea of the equations occurring here we mention the following. If A has just two pairs of points +xl, f x 2 on R and these points do not belong to Z, then det ((XI)', x2) = det (xl, (x')') = 0 (this is the tac-line condition of lemma 27.1). If, however, A has just three pairs k x ' , f x 2 , +x3 on R and these points do not belong to Z, then one has an equation of the form x 3 = u ~ ' ( w , x ' + w 2 x 2 )E R and one further equation det (x2, x3) * det
(XI, (x')')
- det ((x3)',
+det ((x')', (x')')
(XI)')+
det
((XI)', x3)
det (x3, x') = 0.
29.2. Next, we consider asymmetric star bodies in the plane.
5 29
THE CRITICAL DETERMINANT OF A TWO-DIMENSIONAL STAR BODY
333
Let H be a bounded star body, which contains a neighbourhood of the origin and is not necessarily o-symmetric. Then S = H u (- H ) is symmetric with respect to 0,and also a star bJdy. Furthermore, S and H have the same set of admissible lattices. Hence,
d(H) = @),
(6)
whereas H and S have the same set of critical lattices. This reduces the present case to the previous one. Mahler [29c] deals with the particular case that H i s convex. This case gives rise to the following considerations. Let H be a closed bounded convex domain containing a neighbourhood of 0. Suppose that H is strictly convex and that the boundaries of H and - H have at most a finite number of points in common. Let R denote the boundary of S = H u (-If). Put
R+=RnH, R-=Rn(-H),
C=R,nR-.
The following theorem tells us how we can determine the critical determinant and the critical lattices of H.
Theorem 2. Let H.furfi11 the conditions stated and let A be a critical lattice. Then, i f A is singular, allpoints of A n R belong to C ; i f A is regular, then R , (and similarly R-) contains exactly three points x', x2, x 3 of A ; furthermore, for suitable choices of the signs, x3 = +x'+x2.
Proof. In virtue of our hypotheses, a tac-line to S = H u ( - H ) at a point x ECR \ has no point # x in common with S. Hence, by lemma 27.1, it does not belong to any singular critical lattice. Now let A be a regular critical lattice. Since R, and R- are the reflections of each other into the origin, it has at least three points x', x2,x3 on R+ . Furthermore, it follows from our hypotheses and from the theorem on lattice triangles (sec. 3.3) that each two points of A n R , constitute a basis of A. Therefore, x 3 has the form x3 = f x ' + x 2 ; moreover, R+ does not contain more than three points of A . This completes the proof of the theorem. A simple example is afforded by the circular disc H I defined by (7)
( x , -a)2
+(X,i2
5 p2,
One finds that A (H,) = 3(p2 - a')*
0 < CI < p.
+ +
(2a (a2 3p2)*). There is a unique
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CH.
5
critical lattice; it contains the points of intersection of the boundaries of H I , - H I . Mahler [29d] studies in detail the star body S which is the union of two o-symmetric ellipses (8)
a i ( x , ) 2 + 2 P i x l ~ 2 + ~ i ( ~52 )12
( i = 1,2)
2 which both have area 71 (so that ct,y,-fl: = a,y,-Pz = 1). By applying a suitable linear transformation, of determinant 1, we can transform the two ellipses into the special ellipses E l , E , given by
(8')
(X1)Z+(X2)2
5 1,
B
- (x1)2+a--'
(x,),
r 1,
where a is some positive number. The transformation meant does not affect the simultaneous invariant (9)
J =
%YZ--2PlPz+azrl*
Therefore, we have J = a+o-'. The problem is to compute A(El u E,), as a function of J. In doing so we ma] suppose that the ellipses are not identical, so that B # 1 and J > 2. We may even suppose that B > 1 We prove
Theorem 3. The critical lattices of El u E, all belong to the set 9 oj lattices A possessing the following properties: 1". +J3 4 d ( A ) 5 1 2". A has bases {a, b ) , {c, d } with a, b E R 1 ,c, dE R,, where R idenotes the boundary of Ei (i = 1,2). Furthermore, each lattice in 9 is admissible for El v E,.
Proof. A lattice A which has determinant d(A) 2 3J3 and has two independent points on the boundary of El is generated by these two points and is admissible for El (confer the proof of lemma 22.1). A similar property holds for E, . Hence, each lattice A E Y is admissible for El u E,. Now let A be critical for El u E,. Then A is admissible for E l , and so d(A) 2 )J3. First suppose that A is singular. Then, by the argument used in the proof of theorem 2, its points on the boundary are found among the points of intersection of R , and R 2 . There are just four such points, and, by the theorem on lattice triangles, A is generated by these points.
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THE CRITICAL DETERMINANT OF A TWO-DIMENSIONAL STAR BODY
335
Next, suppose that A is regular. Then it has a t least 6 points on the boundary of El u E,. If there are 6 on the boundary R, of E, , then A is critical for El , and so d(A) = 3J3, so that A is critical for E,, too. If exactly 4 points of A lie on R1 and exactly 2 on R,,then, by a suitable small rotation (leaving invariant El), we get a lattice A' which is admissible for El v E, and which has just four points on R1 and no points on R,. By the foregoing, A' is not a singular lattice. Thus X is not critical, against hypothesis. Thus, in all cases, at least 4 points of A belong to R,. By the theorem on lattice triangles, two independent points among them constitute a basis of A . The same property holds for the points of A on R,. It is clear that a lattice A possessing property 2" has determinant d(A) 5 1. The proof of the theorem is now complete. For the points a, b, c, d appearing in 2' we have relations of the type c =ula+uzb,
d =ula+u,b
( u 1 , u ~ , u 1 , uintegers). z
Mahler derives estimates for IuJ, luil ( i = 1,2) and determines the set of lattices A satisfying 1" and 2", for 2 < J < 25. Thus A(E, u E,) is known for 2 < J < 25. For J + co,A(E, v E,) tends to J3.
+
29.3. Mordell [29a, 29b] has developed a method which applies to not too restricted a class of two-dimensional star bodies S a n d which, in many cases, leads to the exact value of A(S). The method is not restricted to bounded star bodies and can be described as follows. Let S be a two-dimensional star body, with distance function f and with a boundary R consisting of four arcs R,,. . ., R4,such that the following requirements are fulfilled: 1". S is symmetric with respect to the coordinate axes and the bisectrices
thereoj: 2". R,lies in the i-th quadrant and terminates on the coordinate axes or has the axes as asymptotes, for i = 1, . . ., 4. 3". Ri is a conuex curve,for i = 1 , . . ., 4. Some consequences of these requirements are as follows. The distance function f satisfies
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CH.
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5
on account of 1". If x = (xl, x 2 ) is a point of R , , then (.I
9
x2+4
4s
(x,+u, x2-c) E S
for u
=- 0,
if x1
2 x2 2 u > 0.
This implies that f(1,u) is steadily increasing for u > 0 and that f (1 u, 1- u ) is non-increasing for 0 5 u 5 1. Consequently,the function g given by g(u) = f ( 1 u, 1 -). - f ( k 0)
+
+
is steadily decreasing for 0 5 u 5 1. We also have
do) = f ( L 1 ) - - f ( 1 ,0) > 0 s(3) =f(% 4) > 0 4)-f(19
g ( 1 ) = f ( 2 , O )- f ( L 1 ) Hence, there is a real number 9 with 3 are two positive numbers a, j? with
(11)
a2+B2 = I,
$01
s 0.
-= 9 5 1, g ( 9 ) = 0. Hence, there
< /I
a,
g(/I/a) = 0.
The last relation (1 1) is equivalent with For p > 0, the star body p S also satisfies the requirements lo, 2", 3". It has distance function pf, but leads to the same pair of numbers E, p. Therefore, it is no loss of generality to suppose that
(13)
f ( a , B ) = 1.
Now consider the 16 points fa', fb', given by
(14)
a' = (a-B, a+B), a3 = (a, B),
a' = (a+/?, -(a-B)), a4 = (B, -a),
b' = (a+B, a-B), b3 = (B, a),
b2 = (-(a-B), a+B), b4 = ( - a , B).
By (12) and (13), all these points belong to R.The points +a', +a2 and the points f a 3 , +a4 are the vertices and the mid-points, respectively, of the sides of a certain parallelogram P,,and they generate a certain lattice A , . Similarly, the points fb' determine a parallelogram Pz and
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THE CRITICAL DETERMINANT OF A TWO-DIMENSIONAL STAR BODY
337
a littice A , . These lattices have determinant 1, and the parallelograms P , , P , have area 4. Resuming we may say that the requirements 1"-3" lead to the existence of two configurations of 8 points all lying on the boundary, such that the points of each configuration are the vertices and the mid-points of the sides of a certain parallelogram Piand so determine a certain lattice A , ( i = 1,2); if the normalization condition (13) holds, then these lattices have determinant 1. For a large class of domains S, the lattices A , , A , are admissible. If it happens that there are no S-admissible lattices of smaller determinant, then A , , A , are critical lattices and so d(S) is known. Mordell expresses the last condition in a manageable form. In this way he obtains the following Theorem 4. Let S be a star body satisfying the requirements lo,2", 3". Let a, p > 0 be determined by (12) and suppose that (13) holds. Denote by RT the sub-arc of R, with end-points a3 = (a, p), b3 = (p, E ) . Then the following statements are true. a) For each point x E RT , there are two points y , z on R,, such that y - z = x. b) If the parallelogram oxyz has area 2 1, for each point x E R:, then A ( S ) 2 I ;if in addition, the lattices A , ,A , introducedabove are admissible for S, then A ( S ) = 1 and A , , A , are critical lattices. We do not give the proof of this theorem which is of an elementary character, but rather lengthy. The main idea of the proof is to apply Minkowski's theorem to an arbitrary S-admissible lattice of determinant 1 and each of the parallelograms Pi considered above and to consider suitable linear combinations of the lattice points thus obtained. The applicability of the theorem is, of course, restricted by the fact that we are looking for critical lattices which have a given configuration of points on the bmndary of S. In some instances it may be advantageous to work with different configurations (see the paper by Clarke cited below). The method may be applied to the star body Ix1x21S 1 (which will be handled, in a different way, in sec. 41). Mordell applies his method to several other domains. We state some of his results. (See also Cassels
[GNI.)
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I. Circular quadrilateral S1, with distance function fl(X1
9
x2)
= I ~ l I + l ~ s l + { ( P 2 - 1 ) ( ( ~ 1 ) 2 + ( ~ 2 ) 2 ) + 2 ~ 1 ~ 2 }(1 *
Critical determinant A($,) critical lattices.
=
s P
{p2+2+2(--p4+ I2p2-4)+)-'.
Just two
11. Star-shaped octagon S, , with distance function fi(X1, x2) =
min (IX,l+PlX,l,
(3-*
IX21+PIXlI)
5P
s 1).
Critical determinant d(S,) = (5-4p+p2)(1 + 2 p - p 2 ) - , . Just two critical lattices if 3-* < p 5 1; four if p = 3-*. Clarke [29a] extends Mordell's method and determines d(S,) for 2-3i I_ p I_ 3*. Similar methods, for other classes of domains, were developed by Bambah [29a] who investigated star bodies in R2 with hexagonal symmetry and by Mordell[29c] who treated star bodies resembling the domain l(x1)3+(x2)31 5 1. 36). Some special two-dimensional domains
30.1. We list several separate results. I. The union S,(a, B ) of the four circular discs (x1)2+(x,)2+ctxl
5 0,
(XJ2+(XZ)
2
+Px2
50
(a
> P > 0).
Critical determinant d(Sl(a, p)) = ct3/3(a2+ P 2 ) - l . Just two critical lattices, each of which is generated by the point (a, 0) and an end-point of one of the four circular arcs which make up the boundary of ,",(a, /I). (Ollerenshaw [30a].)
11. The quadrifoil S2(a) which is the union of the four discs (Xl+a)Z+(x2)2
5 1,
(X1)2+(X,fa)2
61
(0 < ct 6 1).
Here, for 5-* 5 ct 5 1, there is just one critical lattice, generated by the two pairs of points separating the four circular arcs of the boundary. For 0 < a < 5-*, there are two regular critical lattices each containing one of the pairs of points just meant. (Ap Simon [30a].)
I 30
339
SOME SPECIAL TWO-DIMENSIONAL DOMAINS
111. The union S3(a)of the three domains
(x1)z+(x2)z 6 az,
(X1)z+(Xz)zfX1
50
(0
< a < 1).
One has d ( S 3 ( a ) )= max { a ( l -az)*, 2a3(l - az)*, 3a2J3}. (Ollerenshaw [30bi.)
IV. ThestarbodyS,(a)={x: Jxlxz151, l ( ~ ~ ) ~ - ( S x 2~a)}~(O
+
V. The star body S,(a) = {x: Ix1x21 S 1, (x1)z+(xz)2S a } ( a > 0). One has d(S,(a)) = +aJ3, (2a- I)*, 2, (2a-4)*, J5, successively in the a-intervals 0 < a S 2, 2 6 a S 3, 3 S a 5 4, 4 S a =< 9, c1 2 4. (Ollerenshaw [30d].) VI. The star body &(a) = {x: - 1 S x1xz 5 a, Ixl + xzl 5 (a’+ ) ’ . 4 ( a > 0). Critical determinant d(S,(a)) = ( a ’ + 4 ~ ~ ) Two ~ . continuous sets of critical lattices; each lattice from any of these sets has a cell entirely contained in &(a). (Cassels [30a]; confer secs. 27.4 and 45.6).
VII. The star body S7(a) = {x: lxll S 1, 1x21 S l + a x l * (1-21x11)}
(0 < c1 5 3). Critical determinant d(S7(a)) = 1. Same remark as in the previous case. (Cassels [30a].)
30.2. There are also results for domains which are not star bodies.
V I E The domain &(a) = {x: Ixlxzl 1, xz 2 a> ( a 2 0). Critical determinant d(S,(a)) = J 5 , for all a 1 0. The union of S8(a) and the strip 0 S x1 5 a has critical determinant 3(3 + J 5 ) , for all a > 0.
The first result follows from the theory of sec. 28; the second assertion is based on a result of Prasad [45a] concerning the approximation of irrational numbers by rationals. (Mahler [30a].)
IX. The square frame & ( a )
= (x: a
5 max(Ixll, Ixzl) S I} (O<
a< 1).
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CH.
Critical determinant d(S,(ci)) = m-’, where rn = [(1- a ) - ‘ ] . 4 S ci c 1, there is only one critical lattice. (Ollerenshaw [30e].)
5
For
31. The critical determinant of an n-dimensional domain
31.1. In this section we develop, first of all, Minkowski’s method of evaluating the critical determinant of a convex body in R 3 (Minkowski [31a]). We begin by studying lattice octahedra. Let A be a lattice in R 3 and let P be an octahedron with centre a t 0. If the 6 vertices of P are points of A , but no other points # o of A belong to the interior or the boundary of P, then P is called a lattice octahedron (with respect to the lattice A ) . The following theorem says that there are two different types of lattice octahedra. Theorem 1. Let P be a lattice octahedron, with respect to a given lattice A , andlet fx i ( i = 1 , 2 , 3 ) be the vertices ofP. Then A has a basis {a’, a’, a 3 ) , such that one of the following two possibilities arises: I. x i = a ‘ f o r i = I, 2 , 3 11. x i = a’for i = 1 , 2 a n d x 3 = a1+a2+2a3. We shall say that P is of the type I or 11, according to whether case I or case I1 occurs. In the last case, a‘ = x l , a2 = x 2 , a3 = +(- x l - x 2 + x 3 ) , so that A consists of all points x of the form with u 1 = u2
x =~(U1X1+UzX2+U3X3),
(1)
= u3 (mod 2).
Proof of theorem 1. By the corollary to theorem 3.3, A has a basis (a‘, a’, a 3 ) ,such that the three vertices x l , x z , x 3 of P have the following form: x1 = U
l d ,
x2 = v l a1 +uza 2, x3 = w 1 a 1 + w 2 a 2 + w 3 a 3 ,
where u1
> 0, 0 5
Vl
c
212,
0
5
w1
< w 3 , 0 2 w2 c w 3 .
Now, since P is a lattice octahedron, the triangle ox1x2does not contain other points of A. Hence, by theorem 3.4, u1 = v 2 = 1, and so o1 = 0. Therefore, it is permitted to interchange al, a’; so we may suppose that w1 w2, It is convenient to write now w1 = p , w2 = q, w3 = r. We apply the linear transformation B determined by Ba’ = e’ (i = 1 , 2 , 3). We have
s
§ 31
THE CRITICAL DETERMINANT OF AN n-DIMENSIONAL DOMAIN
341
BA = Y, while BP is the octahedron with vertices f(1, 0, 0), f(0, 1, 0), f( p , q, r ) , so that BP consists of the points x satisfying (2) Here, p , q, r are integers with 0 S p S q < r, whereas ( 2 ) is not fulfilled for any lattice point u # f(1,0,0), f(0, 1,0), f.( p , q, r ) . We wish to prove that ( p , q, r ) = (0, 0, 1) or (1, 1,2). To this end, we distinguish three cases. (a) r = 1. Then, necessarily, p = q = 0. (b) r arz odd integer > 1. Then (2) is satisfied by x = ( k , , k, , l), where k, = 0 or 1, according to whether p < +r or p > +r, k, = 0 or 1, according to whether q < +r or q > +r. (c) r an even integer > 1. Put r = 2s. If not p = q = s, then (2) holds for some lattice point x = (k,,k , , l), k, = 0 or 1, k, = 0 or 1. If p = q = s a n d s > 1, then (2) holds f o r x = (1, 1,2). Onlyifp = q = s = 1, there is no new integral solution to (2). Summarizing the results we find that BP, and therefore P, has the required form.
31.2. Now let K be a bounded o-symmetric convex body in R 3 , with distance function$ Let A be a K-critical lattice. Then, by theorem 26.1, there are three independent pairs of points of A on the boundary of K. They determine an octahedron P. It may be that P is not a lattice octahedron. But P certainly contains such an octahedron, and the vertices of this new octahedron also lie on the boundary of K, because K is convex and A is admissible for K. We may conclude that, for each K-critical lattice, there exists a lattice octahedron, with vertices on the boundary of K. We deduce a converse to this assertion, under the condition that a certain finite set of points of A do not lie inside K. Theorem 2. Let P be a lattice octahedron, with respect to a given lattice A , and with vertices x i (i = 1,2,3) on the boundary of K. Then A is admissiblefor K if either 1". PisofthetypeIandf(x) 2 1f o r a l l p o i n t s x = u , x ' + u , x 2 + u , x 3 ~ A ,
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+
CH.
5
+
where one of the ui is 0, k 1, 2 and the other ones are equal to 1, or 2". P i s o f t h e t y p e I I a n d f ( x ) 2 1forallpointsx = + ( + x ' + x 2 f x 3 ) . Proof. First suppose that 1" holds. It is no loss of generality to take x i = e' ( i = 1,2, 3). Then A = Y and f ( e i ) = 1 (i = 1, 2, 3); furthermore, f ( u ) 2 1 for the points u = ( u l , u 2 , u3) specified under 1". Assume that there exists a lattice point u # o with f ( u ) < 1 . For reasons of symmetry, we may suppose that 0 5 ui 5 u2 5 u3;by our assumptions, we then have u3 2 2 and u # (I, 1,2). It follows now from the proof of theorem 1 (see cases (b) and (c)), that the octahedron P,,with vertices + e l , k e 2 , k u contains some lattice point 0 = (ul, u 2 , 1) # + e l , f e z , +u, with v i = 0 or 1 ( i = 1, 2). Since f is convex and since f ( e l ) = f ( e 2 ) = 1 and f ( u ) < 1, we havef(0) < 1. This contradicts our assumptions. Thus f ( u ) 2 1 for all points u # o of Y , i.e., the lattice A = Y is admissible for K . Next, suppose that 2" holds. Again take x i = e' ( i = 1 , 2 , 3 ) . Then A is the lattice of points +u = +(u,, u 2 , u3) with u1 = u2 = u3 (mod 2) (so that A # Y ) ;further, f ( e i ) = 1 (i = 1,2,3) and f ( x ) 2 1 for the 8 points (++, +$, A+). From the relations f(O,O, 1) = 1, f(1, 1, 1) 2 2 it follows that f(1, 1 , O ) 2 1 , f(1, 1,2) 2 1. The last two relations remain true if the coordinates are permuted or one or more signs are changed. Thusf fulfills the assumptions of the previous case (at the points of Y , not at those of A ) . Therefore, f ( u ) 2 1 for all points u # o of Y. Now assume that f ( u l ++, u2++, u 3 + + )< 1 for some triple of integers u l , u 2 , u 3 . We may suppose that 0 5 u1 5 u2 5 u3 and that u3 2 1 . Then we have f ( $ , +,4) < 1, because (+,+,3) is a point of the octahedron with vertices + e l , + e 2 , k ( u , + $ , u2+$, u3++), contrary to our hypotheses. This contradiction proves that f ( x ) 2 1 for all points x # o of A, and so completes the proof of the theorem. The next theorem gives the possible configurations of the points of a critical lattice on the boundary of K .
Theorem 3. There exists a K-critical lattice A with a basis ( x , y , z ] , such that one of the followiizg two possibilities occurs:
I) the 12 points + x , + y , +z, + ( x - y ) , + ( y - z ) ,
k ( z - x ) lie on theboundary ofKandthepoints i . ( - x + y + z ) , + ( x - y + z ) , + ( x + y - z ) lie outside K ,
9 31
THE CRITICAL DETERMINANT OF A N n-DIMENSIONAL DOMAIN
343
11) the 12 points f x , + y , fz, f( x + y ) , f( y + z ) , f( z + x ) lie on the boundary of K and t ( x + y + z ) lie on the boundary of K or outside K. Moreover, if K is strictly convex, then each K-critical lattice possesses this propertv.
Proof. We consider an arbitrary K-critical lattice A , and distinguish two cases. By R we denote the boundary of K. Case 1. Each triple of independent points of A n R determines a lattice octahedron of the type I. Let {XI, x2, x 3 } be such a triple, and let Tl(A) denote the set of the 26 points x = u1x1 u2x2 u3x 3 , where
+
u i = 0 or f 1
+
( i = 1,2,3)
and (ul, u 2 , u3) # (0,0,O).
We observe that the points +2x' f x 2 f x 3 do not belong to R (and therefore lie outside K ) , because the octahedron determined by such a point and the points x 2 ,x 3 is of the type 11. Similarly, the points + x l 5 2x2 fx 3 and the points + x ' + x 2 + 2 x 3 do not belong to R. Therefore, by theorem 2, the lattice A remains K-admissible, if it is subjected to sufficiently small variations such that the points x i remain on the boundary and the set T l ( A )remains admissible for K.This enables us to prove the following assertion (a) i f x , y , z are three points of A n R, then it is not true that eachpoint of T l ( A )n R is of the form
(3)
+z
or
+(x+v,z)
or
t-(y+u,z)
(01,02
integers).
Indeed, suppose that each point of T , ( A ) n R is one of the points (3). We consider two types of variations of A : a) x, z invariant; y varies on R in such a way that y+z, y - z do not become inner points of K,
p ) y , z invariant; x varies on R in such a way that x + z , x - z do not become inner points of K. Under such variations of A , all points of the form (3) remain outside the interior of K, on account of the convexity of K. Now it is always possible to perform an arbitrarily small variation of the type a ) or the type or, if necessary, a combination of such variations, such that A is transformed-into a lattice A' of smaller determinant. But, by our
a)
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SOME METHODS
CH.
5
assumption on Tl(A), the set T,(A) remains admissible under such variations. The same is true for A , if the variations are sufficiently small. This contradicts the hypothesis that A is a critical lattice and therefore proves the assertion. From the assertion (a) and the form of T l ( A ) it easily follows that there exists a triple of points x,y , x E T l ( A ) , such that x,y , z,x-y, y - z , z-x all belong to Tl(A) n R. None of the points -x+y+z, x-y+z, x+y-z belongs to R, because each of these points, together with two of the points x, y , z, x-y, y -z,z-x,leads to a lattice octahedron of the type 11. This proves that, in case 1, always the possibility I) occurs.
Case 2. There are three independent points in A n R determining a lattice octahedron of the type 11. Let xl,x2,x3E A n R have this property and let T,(A) denote the set of points f+xl& +x2f+x3.By theorem 2, the lattice A remains admissible for K, if we vary xl, x2,x3 on R in such a way that no point of T2(A) enters into the interior of K. Such a variation can not lead to a lattice of smaller determinant, because we start with a K-critical lattice. We consider three different variations. a ) Suppose that T2(A)n R is empty. We keep xl,x2 fixed and vary x3 on R in such a way that d(A) does not increase, until some point of T2(A)appears on R. Without loss of generality we may suppose that f(+XI +tx2++x3) become points of R.
p) Put x = xl,y = +x1++x2++x3,z = -x2 and suppose that the points of T 2 ( A )n R are all of the form fz, +x3 or klx+kzy(kl, k2 integral). We keep x,y fixed and vary the pair z,x3 = z+ (2y-x) on R in such a way that d ( A ) does not increase, until some new point of T z ( A ) appears on R.After a suitable permutation of the xi,we get the situation that the points f(f+x' ++x2++x3) all lie on R. y ) Define x, y , z as above and suppose that T2(A)n R contains the points + y , j-(y-x), but no other points. By a similar transformation as under 8) we obtain a situation in which T2(A)n R contains still more points, e.g. f(+x' -+x2+3x3). The remaining pair, viz. f(3x' t+x2-+x3),may lie on the boundary of K or outside K.
It follows from these considerations that, if necessary, we can transform A into another K-critical lattice (again denoted by A ) with the property that Tz(A)n R contains the points
9: 31
THE CRITICAL DETERMINANT OF A N n-DIMENSIONAL DOMAIN
345
whereas k 4('. +(3s2-+x3) lie on the boundary of K or outside K. If K is strictly convex, then our considerations tell us that the original lattice has already this property. The assertions of the theorem now follow on taking x = -1x1 + + x 2 + + x 3 , y = +x1-12x 2 + z1x 3, z = -*x1-+xZ-J$. 2 Remark. We observe that only in case 2 it is necessary to pick out a particular critical lattice. Presumably, the assertion of theorem 3 is generally true for all K-critical lattices, but this has never been proved. 31.3. The foregoing results can be used to evaluate, at least theoretically, the critical determinant of a given convex body in R 3 . Let K be a bounded o-symmetric convex body in R 3 , with boundary R. By the foregoing, there exists a K-critical lattice A , with a basis {x, y, z}, such that one of the following two possibilities arises: 1) each triple of independent points of A n R determines a lattice octahedron of the type I and, moreover, property I of theorem 3 holds, 2) property I1 of theorem 3 holds. We denote by Z the set of points of A n R enumerated in theorem 3 under I and 11, respectively. Thus Z consists of k pairs of points, where k = 6 in case 1) and k = 6 or 7 in case 2). Furthermore, as follows from the proof of theorem 3, there are three independent points xl, x 2 , x 3 E Z such that, in case I), each point x having the form * k x 1 k x 2 + x 3 or fx'+kx2+x3 or + x ' - t x 2 + _ k x 3 (k = 0 , l or 2) and not belonging to Z, lies outside K, whereas, in case 2), each point x $ Z of the form t- +xlk+x2 f x 3 lies outside K. Now let f x l , . . ., f x k be the complete set of points in Z and let H I , . . ., Hk be half-spaces not meeting the interior of K, such that xi lies on the boundary of H i , for i = 1, . . .,k. The fact that the points +XI, . . ., * x k lie on R yields k equations for the 9 coordinates of the points xl,x2, x 3 . To get more equations we consider small variations of A which are such that x i remains a point of Hi, for i = 1, . . ., k. By theorem 2 and the foregoing remarks on the set 1,the lattice A remains admissible for K, and so d(A)does not decrease, if the variation performed is sufficiently small. Thus ldet (xl,x2, x3)/ does not decrease if xl,x 2 , x3 are subjected to small variations not affecting certain k linear inequalities. This leads to 9 -k further equations for the coordinates of
*
346
cn. 5
SOME METHODS
xl, x2, x3. So, in the aggregate, there are 9 equations for these coordinates. Among the solutions of these equations there is a set {xl, x2, x3} de-
termining a K-critical lattice.
Minkowski sets up and solves the equations meant in the case of the octahedron Po given by (4)
Ix,l+lx2l+lx3l
5 1.
He finds that there are 8 critical lattices. They are transformed into each other by reflections into coordinate planes; one of them is the lattice generated by the three points al
=(-”6 ,
1 2) 62 6
u2
=(? 67
-2
1)
69 6 7
u3
= (L 6, 3 69 2 6 ) 9
so that (5)
d(P,)
=
det (u’, u 2 , u3) = -&$.
Accordingly, the density of densest lattice packing (sec. 20) of Po is given by
(6)
b(P0) = V(P,)/d(2P,)
= +$.
Whitworth [31a, 31b] dealt with the domains
lxil 5 1 (i = 1,2,3),
Ix1+x2+x3l
5 z (0 < z < 3);
5
{ ( X I ) ~ + ( X ~ ) ~ } ~ + I X ~ 1. I
The first domain is affinely equivalent to Po if z = 1. For both domains the numerical details of the procedure are rather involved. Minkowski [31a] also stated that the density of densest lattice packing of an arbitrary tetrahedron P I is equal to +3(Po) = 2K. This statement was shown to be wrong by Groemer [31a] who found that 8 ( P l ) 2 > 4%. Some authors have tried to generalize the foregoing theory to the four-dimensional case. Thus Wolff [3la] continuing work by Brunngraber derived four-dimensional analogues of theorems 1 and 2 and used the results to obtain the critical lattices and the critical determinant of the unit sphere in R4.Mordell [31a] gave a simple proof of the four-dimensional analogue of theorem 1. 31.4. The theory of sec. 29.1 can, to a certain extent, be generalized for n-dimensional domains. In particular, we consider convex bodies and
ri 31
THE CRITICAL DETERMINANT OF A N n-DIMENSIONAL DOMAIN
341
derive two finiteness theorems due to Cohn [31a]. For p > 0, we denote by C,, the sphere 1x1 S p . The volume of the unit sphere in R" is denoted by IC, (m 2 1).
Theorem 4. Let p be a positive number and let K be a (closed) bounded o-symmetric convex body in R" containing C,. Suppose that K does not contain a lattice point u # o with lul 5 cc,p'-", where u, = n 2 " - ' ~ ~ - ' ~ . Then Y is admissible for K (so that A(C,,) = p"A(C,) 1). Proof. Suppose that K contains a point x with 1x1 = a,pl-". Denote by H, the,(n- 1)-dimensional linear subspace of R" which is orthogonal to the vector x , and by K, the convex hull of f x and the (n- 1)-dimensional sphere C , n H,. The set K, is an o-symmetric convex body contained in K. It has volume
So it contains a lattice point u # 0 . This contradicts the hypotheses of the theorem. Hence, Kdoes not contain a point 1x1 with 1x1 = ~ l , , p ~ and - " so K i s contained in the sphere 1x1 < q,pl-". Applying again the hypotheses of the theorem we may conclude that Y is admissible for K. Theorem 4 can be applied as follows. Let K be a bounded o-symmetric convex body in R" and let A be a lattice having n independent points x l , . . ., x" on the boundary of K. By the corollary to theorem 3.3, there exists a basis {a1,. . ., a"} of A such that, for i = 1 , . . ., n, x i has the form xi = u l i a ' + . +uiiai, where uli,. . ., uii satisfy (7)
05
uki
< uii
(k = 1 , .
.y
i-1).
We consider the linear transformation A given by Ae' = a' (i = 1, . . ., n ) and put K' = A-'K. Then K = AK' and x i = Au', where (8)
i
u = (Uli,.
. ., u i i , 0,.. .)0)
( i = 1,.
. .)n).
The points d,. . ., u" all lie on the boundary of K'. Hence, K' contains the generalized octahedron P with vertices f ul, . . ., ku". Now suppose that A is admissible for K. Then X I , . . ., x" are a set of minimum vectors with respect to K . By (9.13) the index ldet (uki)l of this
348
CH. 5
SOME METHODS
set of vectors in A satisfies the estimate
-
ldet (uki)l = u l l u 2 2 * * u,, S n ! . (9) Clearly, there are only finitely many generalized octahedra P, with vertices fd,. . ., fu", such that the points uihave the form (8) and have coordinates u k i satisfying (7) and (9). The intersection of these octahedra contains a sphere C, , where p = pn is a positive constant only depending on n. Applying theorem 4 we obtain Theorem 5. The critical lattices of a bounded o-symmetric conuex body K, with distancefunctionf, are found among the lattices A having n indepen+ u i i a i on the boundary of K, where dent points xi = uliai+ {a', . . ., a"} is a basis of A and where the coeficients uki satisfy (7), ( 9 ) and
-
f(u,al+
(10)
for all u
= (ul,
. . ., u,)
. - .+u,an) 2 1
with 101 5 ct,,p;-"
=
n2"-i~n-lp~-".
By considering special lattices (admissible for the unbounded star body Ixll{max (IxzI,. . ., I x , , ~ ) ) "5- ~l), Cohn shows that theorem 4 is no longer true if a, is replaced by a sufficiently small positive constant & or ct,,p1-" by &p-", where p, > 0 and rs
c = (x = (x1 a x2 > xg): (x,, x2) E K , 1x31 5 11. (1) It is trivial that d(C) S d ( K ) . The question arises whether or not in this relation the equality sign holds. In sec. 20 we proved already theorem 20.6 which we restate as Theorem 1. If K is a bounded o-symmetric convex domain in R2 and C is deJined by (l), then d(C) = d ( K ) .
We mention some other proofs of theorem 1. We denote the boundary of C by R and the part of R in the plane x , = 1 by R,.
9: 32
349
SOME SPECIAL DOMAINS
Yeh [32a] deduced theorem 1 by using Minkowski’s theorem 31.3. Chalk and Rogers [32a] deduced it from the following two propositions: (a) There exists a C-critical lattice A , which has three points a, b, c on R, such that, in the plane X, = 1, c is an innerpoint of R , . (b) Let a‘, b’, c‘, be three points of K not lying on a straight line. Then the lattice A ; , with a basis (6’-a‘, c‘-a‘}, is a covering lattice of K. r f , in addition, one of the points a‘, b‘, c‘, is an inner point of K, then Ah is a covering lattice of int K. Proposition (a) is proved by taking an arbitrary C-critical lattice A and by subjecting it to a suitable transformation of the type xi = x1,x; = x2, x i = E X , +px2 f x , . Proposition (b) follows from the fact that Ah is a covering lattice of the hexagon P which is the convex hull of the points fa’, icb’, kc’. Theorem 1 can be deduced from these propositions in the following way. It is no loss of generality to suppose that K is strictly convex. Now take the lattice A , provided by (a). Consider an arbitrary point X E A , with I X , ~ < 1 and denote by a’, b’, c’, x’ the projections of a, b, c, X, respectively, onto the plane x j = 0. The three points a’, b‘, c‘ do not lie on a straight line, as otherwise A , would not be admissible for C, and c’ is an inner point of K. Therefore, by (b), there are integers u1, u2 such that
x ’ - u l ( b ’ - a’) -u2(c’- a’) E int K . Now the point x-ul(b-a)-u2(c-a)
x-u,(b-a)-u,(c-a)
has last coordinate x 3 . Thus Eint C.
Therefore, since A, is admissible, this point coincides with the origin. It follows that the points of A , are distributed over the planes x3 = k, k integral, and that A , is generated by a lattice Ah in x3 = 0 and some 2 d ( K ) , so that point in x3 = 1. We may conclude that d(A,) = d(C) 2 d ( K ) . Since we also have d(C) 2 d ( K ) , this proves the theorem.
a(&)
Few [32a] adapted the foregoing proof so as to obtain the following result.
Theorem 2. I f K is a bounded symmetric (but not necessarily o-symmetric) convex body in R2 containing a neighbourhood of o and if C is defined by (I), then d(C) = d ( K ) .
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SOME METHODS
CH
5
The relation A(C) = A(K) also holds if K is the nonconvex star body Ix1x215 1, as was shown by Varnavides [32a]. But this relation is not
generally true. Thus Rogers [32a] found an example, where A(C) < 6 ( K ) . Later on, Davenport and Rogers [32a] constructed a class S of star bodies K such that the quotient d ( C ) / A ( K )takes arbitrarily small values on 2;a domain K in this class is obtained as the complement in the plane of finitely many small angular regions, with vertices at the points (k, 9k,, m),where 9 is a fixed irrational number. Woods [32a] proved that the three-dimensional sphere (xl)' (x,)' +(x3)' 5 1 has the same critical determinant as the corresponding cylinder C in R4.Further, Woods [32b] showed that a convex body K in RZwhich is symmetric with respect to the point o and the line x1 = 0 may not possess a critical lattice A' contained in a three-dimensional K*-critical lattice, where K* is the convex body in R 3 obtained by rotating K about the x,-axis.
+
+
+
32.2. Next we consider spheres. We denote the unit sphere in R" by K,, and deal with a method of Ollerenshaw [32a] to evaluate d ( K 3 ) . Let A be a K3-critical lattice. By a suitable rotation, it is transformed into a K3-critical lattice A , generated by three points a', a', a3 on the boundary of K 3 , where a' is the point (1, 0, 0), a' has the form (cos 9, sin 9, 0), with 0 < 9 1,
a3 has a positive last coordinate c, say. Since A , is admissible, we necessarily have 3. 5 9 5 3. Now let H,, denote the plane x3 = t ~ , and G the circular disc in RZ having on its boundary the points 0 , a', a'. The disc K3 n Ha has radius p = (1 - o ' ) ~ , and G has radius 0' = (2cos @)-I. Suppose that p' < p . Then the planar lattice L, with a basis (a', a'}, has three points 0,d,a' inside some circular disc (not centered at 0) of radius p . Hence, on account of proposition (b) discussed in the foregoing subsection, L is a covering lattice of the interior of this disc. This contradicts the fact that the set A , n H,, = = u 3 + L has no point inside the disc K3 n H a . Therefore, p' 2 p . Hence, CT 2 (1 -(p')'}' = (1 -4(CoS @)-']',
so that
d(A,) = d sin 8 2
( ~ ( C O S$9)'
-I>* sin $8.
0 32
35 1
SOME SPECIAL DOMAINS
The minimum of the last expression on the interval 3n S 9 3. is attained at the point 9 = +n and is equal to 2-+. So we have d(A,) 2 2-*. It is easily verified that, in fact, there exists a K,-admissible lattice of determinant 2-*. Hence, finally, d(K3) = 2-+.
(2)
Ollerenshaw [32b] proves, by a similar method, that (3)
4Kd=
+a
Chalk [32a] and Ap Simon [32a] investigate the domains K3(a) = (x:
(xi)2+(x2)2+(x3)2
5
K;(a) = {x: ( x 1 - a ) 2 + ( x z ) z + ( x 3 ) 2
1 7
1x31
s I},
a>,
respectively, and find that (0 < a < 1)
(4)
d ( K 3 ( a ) ) = 3a(3-a)*
(5)
d(K;(a)) = +(l-a2)(aJ3+(2-a2))~
(0 < a < 1).
The proof of (5) is based on a partial generalization of the theory of sec. 31.3 to bodies of the form K u ( - K ) , where K is a strictly convex body in R 3 containing a neighbourhood of 0 .
32.3. Finally, we consider n-dimensional cubes. The cube W: lxil S 1 ( i = 1, . . ., n) has critical determinant 1; this follows from Minkowski’s theorem and the fact that Y is admissible for W. It is also easy to see that all lattices A having the following property (P) are W-critical:
(P) there exists apermutation of coordinates transforming A into a lattice A’ with a basis
where the coordinates aki (k < i ) are arbitrary real numbers. But it is much more difficult to prove that there are no other Wcritical lattices In fact, this is equivalent with the theorem of Minkowski-
352
SOME METHODS
CH.
5
Haj6s stated in sec. 12.4 (each system of linear forms of the type (12.3) determines a lattice basis of the form ( 6 ) and conversely). Ollerenshaw [32c] proved that the hollow cube
w(u) = {x: ct
max lxil
i = l , ..., n
5 11
(0 < ct < 1)
has critical determinant m-", where m = [(l - cc)-']. With the help of the theorem of Minkowski-Haj6s she showed that m-'Y is the only critical lattice. Ap Simon [32b] investigated the excentric cube W ( a ) given by
(0 < ai < 1 ; i = 1 , . . ., n).
[xi-ail I1
(7)
This cube is connected with an interesting space filling body; therefore, we give some details. By the substitution xi = a i y i (i = 1, . . ., n) the cube W ( a )is transformed into the parallelotope G : lyi-lJ
i = 1, . . ., n).
(bi = a,:';
5 bi
Put H = G u (- G ) and define K as the set of points y with (8)
lyil
2 l+bi,
(i # j ; i, j = 1 , . . ., n).
5 bi+bj-2
lyi-yjl
Then K is an o-symmetric convex body. It is contained in H , and the lattice A , with a basis (b,+l, 1, * ., I), ( l , b 2 + l , . *
. .)l), . . ., (1, 1,. . .,b,+l)
is admissible for G , H a n d K. Furthermore, 2-"V(K) = (1+2C(bi-1)-1).n(bi-1) i
i
= d(A0).
These facts imply that K i s space filling and that d ( K ) = d ( H ) = d(G). The lattice A , is the only critical lattice of K, H a n d G. 33. A method of Blichfeldt. Density functions 33.1. In the remaining part of this chapter we deal with some methods
which yield lower bounds for the critical determinants of various types of rz-dimensional bodies. We begin by explaining a method which was used first by Blichfeldt [33a], in the case of n-dimensional spheres. Roughly speaking, the idea is as follows (Rankin [38a]).
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A METHOD OF BLICHFELDT. DENSITY FUNCTIONS
353
Let K,, denote the unit sphere in R" and let 6; = 6*(Kn)be the density of densest packing of translates K,+x (sec. 20.3). Consider an arbitrary packing of translates of K,. Replace the spheres in this packing by larger spheres (possibly overlapping) of fixed radius po > 1 and fill each of them with a certain amount of mass, of variable density, such that the total mass at an arbitrary point of space does not exceed 1. Then the product of the average number of spheres per volume unit and the mass of a single sphere will not exceed 1. This leads to an upper bound for :S and so to a lower bound for A(&). A precise formulation of this idea is given by
Theorem 1. Let po be a positive number and let 6 ( p ) be a density function which is non-negative and continuoui f o r 0 _r p 6 po and which vanishe$ j o r p > po. Suppose that, for each packing ( K n + x ' } ~ = = l , m
(1)
CS(l~-x'l)
1
r= 1
for all X E R " .
Then S,* 5 I-', where
Remark. The sum appearing in (1) is finite, because S(lx-xrl) = 0 if Ix-x' 1 > p o . The density of an enlarged sphere poK,+Y at a point x only depends on the distance Ix - x'l . Proof of theorem 1. Let W denote the cube lxil <= 1 (i = 1,. . ., n). Consider an arbitrary packing {K,+x') (r = 1,2, . . .) and, for z > 0, denote by n, the number of spheres K, +x' which are entirely contained in the cube z W , we may suppose that the centres of these spheres are just given by x', . . ., xnz. The concentric spheres poKn+X' ( r = 1,. . ., n,) are contained in (z+po- 1)W c ( r f p 0 ) W . Hence, (3)
~(lx-x'l) =
o
for r = I , .
. ., n,,
if x $ ( z + p o ) W .
Using (1) and (3) we find r
m
P
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cn. 5
SOME METHODS
The last integral is equal to &6(p)d(K,,p") = K,,I, where So we have 2"(2+p0)" 2 n,ic,,I.
K,, =
V(K,,).
Hence, the quotient of the total volume of the n, original spheres K,, +xi and the volume of the cube z W containing them satisfies
n,K,,/V(zW) = n,~,,(2z)-" 6 (1 +po/z)"l-'. It follows now from the considerations of sec. 20, in particular theorem 20.5 and the definition of 6: = a*(&), that 6: 5 lim (1+p,,/z)T1 r+ m
= I-'.
This proves the theorem.
In sec. 38 we shall see that theorem 1, with a suitable choice of the function 6, leads to much better estimates for A(&) than the theorem of Minkowski. Here we enunciate a more general theorem (which will be applied in sec. 39). Theorem 2. Let K be a bounded o-symmetric convex body, with distance functionf. Let po ,(r be positive numbers and let 6 ( p ) be a density function which vanishes for p > p o . Suppose that, for each packing {K+x'}:==, ,
(4) Then one has
r
6(f"(x-xr)) 6 1
6*(K) 5 I-',
for all x E R". *
where I = j ) ( p ) d p " / " .
We omit the proof, because it is very similar to that of theorem 1 and because we deduce below a still more general result. 33.2. On account of (20.5) and (20.12), the inequality (5) implies (6)
d ( K ) 2 2-"V(K)Z.
We generalize this relation as follows (compare Rogers [41a]).
Theorem 3. Let S, T be two star bodies, with distance functions f, g, respectively. Suppose that T is bounded. Furthermore, let p, y, p o , IS be
*
For simplicity, we writefc(z) instead of {f(z)}".
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A METHOD O F BLICHFELDT. DENSITY FUNCTIONS
355
positive numbers and let 6 ( p ) ( p 2 0) be a density function vanishing for p > p o . Finally, suppose that the function 6 ( p ) does not exceed y and, more generally, that f o r any rn points x l , . . ., x m with
( 7)
f(xk-x.') 2 p
(k # 1 ; k, 1 = 1 , . . ., m),
one has m
(8)
6(g'(x-xr)) 5 y,
for a11 x
E R".
1
Then A ( S ) satisjies
Proof. If d(S) = coy there is nothing to prove. Suppose that d(S) is finite, and consider an arbitrary lattice A which is admissible for PS. Let P be a fundamental cell of A , let x be an arbitrary point of R" and let x l , . . ., x m be the points of A with g"(x-x') < po (r = 1 , . . ., m). Then (7) is satisfied and, moreover, Stg'(x-y)) = 0 for each point y E A which is not one of the points xl,. . ., xm. So, in virtue of our hypotheses,
2 6(g"(x-y))
YEA
c 6(gu(x-xr)) 6 Y*
m
=
r= 1
Of course, the number m depends on x. Integrating over P we get
The integral in the second member is equal to
Jomd(p)d(V(p'lUT))= V(T)JmS(p)dpni" 0 = IV(T). SinceA was an arbitrary PS-admissiblelattice, the assertion of the theorem follows.
If Sis bounded and convex, then the condition (7) means that the bodies $/?S+xk (k = 1, . . ., m ) do not overlap. Therefore, theorem 3 reduces to theorem 2 (with ( 5 ) replaced by ( 6 ) ) if we suppose S to be convex and take T = S, p = 2, y = 1. A particular case, of special importance
CH. 5
SOME METHODS
356
for the applications, arises if we take (10)
6 ( p ) = max (0, z'-p),
y = yo?',
z a positive constant. In this case, the last condition of theorem 3 (with
g = f) can be formulated as follows: if
points in R" such that
f ( x - x r ) < z (r = I,.
. ., m),
x, x l , . . ., xm are any m+ 1
m
Cf"(x-x') 1
< (m-yo)zu,
thenf(xk-xt) < /Ifor at least one pair of indices k, I with k # 1. One can describe a general situation in which this condition is fulfilled. This is done in the following theorem which is a generalization of theorems of Blichfeldt [38a], v. d. Corput and Schaake [40a] and Hlawka [13a].
Theorem 4. Let S be a bounded star body, with distance function f, and let y o , CT,z be positive numbers with z" > ((n/o)+I)yo{ V ( S ) } - ' . Then, f o r each lattice A with determinant d ( A ) = 1 , there exist a point x E R" and in > yo points XI, . . ., x" E A , such that (11)
f(x-x') < z (r
m
=
1,. . ., m), Cf"(x-x')
< (m-y,)~".
1
Proof. We shall apply the theorem of Blichfeldt to the function q ( x ) = ~(f"(x)) = max (0,z"-f"(x)).
(12)
As in the proof of theorem 3, we have
By hypothesis, the last expression is > yozu. Hence, by Blichfeldt's theorem and the relation d ( A ) = 1, there exists a point x E R" such that q ( x - y ) > yozu. Thus there exists a set of points X I , . . ., x mE A such that
zyPA
q(x-x')
z0
m
(r = 1,
. . ., m), C q(x-x') > yo?'. 1
The terms in the last sum are all z'; therefore, the number of these terms is greater than y o . Using (12) we get the relations (11). Applications will be given in sec. 40.
8 34
357
A METHOD O F BLICHFELDT AND MORDELL
34. A method of Blichfeldt and Mordell 34.1. In his original memoir [5a], Blichfeldt applied the theorem which now bears his name to the problem of the simultaneous approximation to n- 1 given real numbers by fractionspi/pn (i = 1, . . ., n- 1) with equal denominator pn ; in this way, he could ameliorate the result obtained by Minkowski (sec. 45.2). In fact, he applied theorem 16.2 to some special lattice of determinant 1 and a certain star body T with the property that the difference set 9 T is contained in the star body S given by (1)
1x11
. {max(lx,l,. . ., Ix,O]"-
5 7,
where y is a suitably chosen positive constant. Later on, Koksma and Meulenbeld [34a, 34b] and Mullender [34a] applied a similar method to more general approximation problems (sec. 45.2). Mordell [34a] and Mullender [34b] analyzed the underlying principles and deduced some general theorems. These theorems will be dealt with below. We consider pairs of star bodies S, T satisfying B T c S. As a consequence of Blichfeldt's theorem, we have
Theorem 1. Let S, T be star bodies iir R" with 9 T c S. Then A ( S ) 2 V ( T ) . Proof. d(S) 2 A ( 9 T ) 2 V ( T ) ,on account of (17.11) and theorem 17.6. Theorem 1 is the basis for our further considerations. The main problem is to choose T, for given S, in such a way that 9 T c S and V ( T ) is reasonably large. In the following, the star bodies S, T will always be symmetric with respect to the n coordinate planes x i = 0 (1 5 i 5 n); we put (2)
S, =SnO+,
T+ = T n O , ,
where 0, is the hyperoctant xi 2 0 (i = 1, . . ., n). In general, T will be a set in Rn given by a pair of inequalities of the type (3)
fI(x1, *
*
.) x p )
s
70
fZ(xp+l, * *
-3
x n ) 6 $(fi(xi
9
-
* -9
xp)),
where p is a given index with 1 S p 6 n - I , zo is a positive number, is a non-negative and steadily decreasing function of z on the interval 0 5 z 6 zo and fi, fz are the distance functions of a bounded o-symmetric convex body Kl in RP and a bounded o-symmetric con-
$(z)
SOME METHODS
358
CH.5
vex bsdy K2 in R n - p ,respectively. The star b3dy S will be defined by a similar pair of inequalities, with the same indexp. We deduce a formula by which V ( T )is expressed in terms of the functions I),f l , f 2 . Let V , , V2 denote the volumes of K , , K 2 , respectively. Then, for 0 < z 5 r o , the bodyf, S z has volume ol(z) = zpV, and the bDdy f2 S $(z) has volume w2(z) = $ ( T ) ’ - ~ VApproximating ~. T by a .finite number of ‘cylindric’ shells
I fl(Xl? * * < z ( k + I), f 2 ( x p +1 * x,) s (0 = (0) < ~ ( 1 )< * * * < z ( k ) < * - * < z ( f ) = z0),
$4
*)
7
’7
4w))
of volumes ( o , ( ~ ( k +l ) ) - - ~ , ( ~ ( k ) ).}w2(2(k)),we get the result that
34.2. In this subsection we deal with the case n = 2, p = 1. Let S c R2 be given by
s
5 cP(lXll), where z1 is a positive number and q ( ~ (0 ) 5 z 5 7,) is a convex, nonnegative andsteadily decreasing function of z. Let 5 be a fixed number with (5)
0<
(6)
1x11
5 < T,
717
1x21
and let
x2 = p-vx1
( p , v > 0)
be the equation of a tac-line to the curve x2 = q ( x l ) at the point (t, q(t)). We discuss some further conditions to be imposed upon rp and 5. The equation ( 6 ) may be written as
cc1x1+cc2x, = 1,
with a, = p ( - l v , cc2 = p - ’ .
In particular, cc,~+a,cp(<) = 1. Since the r8les of x l , x2 may be interchanged, it follows that it is no loss of generality to suppose that a2rp(5) 2 .), i.e. (7)
4 5 ) 2 +P. Next, let us consider the function q1 given by
9: 34
A METHOD OF BLICHFELDT AND MORDELL
3.59
This function is convex and steadily decreasing, whereas q(0) = p, = 0. Then we have
q l ( z , ) = ~(7,). Suppose that cp(zl)
(9)
cpl(-b,) 6
4F (7), this inequality implies that cp1(3tl)6 q(5); since q(5) = q l ( 5 )
By and pl is steadily decreasing, it follows that $5, 2 5, so that ql(3z1) = = p(&,). I n the following we do not assume that cp(zl) = 0, but only require that
dhl)6 3P.
(10)
We now come to the question of choosing T. Our first thought is to fit into S an o-symmetric convex body K and to take T = 3K. Such a body K can be obtained as follows. Since x2 = p-vx, is the equation of a tac-line and the function cp is convex, we have p-VT
5 q(t)
for 0
5z
z,.
By (lo), this implies that p-+vt, 5 3 p or also p-vz, 6 0. Therefore, the part of that tac-line intercepted by the axes is entirely contained in S. So the rhomb K : vlxll+ Ix2JS p is contained in S. Thus
9 T c S and [herefore A ( S ) 2 V ( T ) , i f T is the rhomb vIxl(+Ix,l 5 $p. It must be remarked, however, that we obtain the same result if we apply directly Minkowski’s theorem to the rhomb K. A less trivial result is obtained if one modifies in an appropriate way the star body T. This was done by Mordell[34a] who proved the following
Theorem 2. Let zI > 0 and let cp(z) be convex, non-negative and steadily decreasingfor 0 5 z zl.Furthermore, let x2 = p-vx, be the equation of a tac-line to the curve x2= q ( x l ) at some point (5, q(5)) with 0 < 5 < z1, and suppose that the inequalities (7) and (10) hold. Finally, let TO 2 5 and $(z) be defined by
Then,
if S
1x11 6 T O ,
V(T)-
is given by lxll 5 zl, lxzl 5 q(lxll) and T is given by 1x21 5 $(lxll), the critical determinant A ( S ) is not less than
360
SOME METHODS
CH.
5
Proof. Let x = ( x l , x z ) , y = (yl, y z ) be two points of T,. Then x,+yl 5 22, S z, since cp(ro) = t p 2 cp(+tl), on account of (11) and (10). We prove that we also have xz y z 5 q ( x 1 y,). It suffices to treat the following three cases.
+
+
5, 0 5 y , 6 5 . Then
Case 1. 0 5 x1 5
5 * ( X l ) + * ( Y l ) = CL--~xl+Yl) 5 c p ( X I + Y l ) . Case 2.0 5 x1 5 ( 5 y 1 5 zo. Then, since cp is convex and the tac-line at the point (t,cp(t))has slope -v, the increment cp(yl+xl)-cp(y,) X,+Y,
exceeds -vxl. Hence, by ( 1 l),
5
X,+Y,
*(Xl)+*(Yl)
Case 3. 5 5 x1 5 zo,
= -vxl+cp(Yl)
5
cp(Xl+Yl).
t 5 y1 5 zo. In this case,
xz + Y Z - cp(x1 + Y J 5 -cL+cp(x1)+cp(Y1)-cp(x1 + Y J
5 -cL+cp(5)+cp(Yl)-cp(s+Yl)
s
-p+cp(t)+v5
5
5
= 0.
We may conclude that x +y E S, , if x, y E T , . Now let x, y be two arbitrary points of T and let a = (a,, a z ) be the point (Ixli I y l l , lxzl + lyzI). By the result just obtained, a E S , . Moreover, z E S if lzil 5 a, (i = l , 2). Hence, x + y E S. This proves that 9 T c s.
+
34.3. Mullender [34b] derives the following n-dimensional analogue. Theorem 3. Let T ~ t,, p, v and the function cp satisfy the same conditions as in theorem 2 and let zo and II/ be defined by (1 1). Next, let p be an index with 1 5 p 5 n- 1, let f l y f z be the distance functions of bounded osymmetric convex bodies (of volumes V , , V,) irl RP, Rn-p, respectively, and let S be the star body in R" given by (12)
fl(X1
5
*
-Y
XP)
I z15
fZ(X,+l,
-
*>
x,) 5
P(fl(X1
9
* '
.Y
x,)).
Then one has
Proof. Let T be defined by ( 3 ) and let xyy be two arbitrary points of T.
9: 34
361
A METHOD OF BLICHFELDT AND MORDELL
Write x = (x’, x”), where x’ = ( x i , . . ., xP), x” = (xp+l,. . ., X n ) , and similarly y = (y’, y”). Proceeding as in the proof of theorem 2 we find that
fh’) +fl(Y’) s
Z1
Y
s cp(.W)+fZ(Y’)).
fi(x”)+fz(Y”)
Since the functionsf, ,fzare convex and cp is decreasing, it follows that
s
fl(X’+Y’)
Z19
fZ(~”+Y”)
Icp(fdX’+Y’))Y
that is x + y E S. So we have 9 T c S. The assertion now follows from theorem 1 and the relation (4). We give two applications of theorem 3. The first one is essentially due to Mordell [34a]; the original result of Blichfeldt [5a] and, according to a remark of Mullender [34a], also a result of Koksma and Meulenbeld [34a] are simple consequences of it (see sec. 45.2). Theorem 4. For 5 > 0, let the star body S,(t) be given by (14)
1x11 (max (IxzI,
lxil
- IxnI)>”-l s 1, s 2{-l’(“-l) (i = 2 , . . ., n).
Then the critical determinant A (Sl(())satisfies 1 n-1 where 6 = 1- (15) A(Sl(t)) > (1+ (1+6), n-1
(
-)
Proof. We apply theorem 3, with
P = 1, fi(x)
= yzl
IXlI,
fi(x2
.
, . ., xn)
= max (Ixzl,
= 2”t, cp(yz) = T-l’(n-1).
We put w = cp(g). The tangent to the curve c = cp(z) at the point
c = p-vyz, Clearly, w 2 *p
(t, w ) is given by
n 1 where p = __ an, v = __ an. n-1 n-1
2 t.0 and so
v, = 2 , v, = 2 n - l ,
. . ., IxnI),
cp(<)
(3
1 +p 2 cp(tzl). Next, we have
n-1
yzo =
2(n-1)
=
(-)
2(n-1)
n-l
=
(--)
n-l
5.
362
CH. 5
SOME METHODS
Hence, by (13) and the definition of $(z) (see (11)),
The first term on the right is equal to
_1 . {p -(p-2v5)"} vn
n-2
n-1 n
= -*
n
the second term is handled by substituting z = (2(1-p)/pY-l found to be equal to pn-l(l-p)-'dp
-
e(-1 n
= (n- 2 ) / ( 2 n - 2 )
+(n - 1)2ns0
n-2 n-1
In the last integral we expand (l-p)-' po = ( n - 2 ) / ( 2 n - 2 ) ,
= - n- P- 1o
p"(1 - p ) - l d p .
into a power series. Writing
p1 = p o ( n + l ) / ( n + 2 ) we find that
p"(l--p)-'dp = (n-1)2"-
and is
1 (n+l
=
1
1 __
pP;+2+
n+2
n+3
p"03+
I
...
=
n+1.
n
n+l
n+l nf2
n+l = 2np;++l
.
n-1 (n+l)(l-d
= (2po)n+'
The last expression is greater than
Collecting the results we find that
+2)(n-1)~ - (n+( r 1l)(n2 + 3n - 2 )
'
9: 34
363
A METHOD OF BLICHFELDT AND MORDELL
This proves (15).
Remark. Minkowski obtained essentially the crude result
One gets this result if one fits into Sl(5) the convex body given by max ( I x , ~ , . . ., Ix,,~) p-v(xll, of volume
Macbeath [34a] applied theorem 3, with p = 1, to a star body S = { x :f ( [ x l l , .. ., Ix,,~) 5 l}, where the distance function f has the following property: f ( x l , . . ., x,,) is steadily increasing in each of the n variables, in the hyperoctant xi 2 0 (i = 1, . . ., n), and the domain of points x with f ( x l , . . ., x n ) 2 1, xi 2 0 ( i = 1 , . . ., n ) is convex. He could easily prove
Theorem 5. Let S be a star body of the type described. Let (t,, . . ., t,,) be apoint on the boundary, with t i > 0 (i = 1, . . ., n), and let Cpiri = 1 be the equation of a tac-plane at this point; suppose that PI t1 5 n-l. Then (16)
d(S) 2
2"(P1p,
- - -/I,,(n-l)!)-'
x
where for each t ,c p ( t ) is the largest number such that the (n- 1)-dimensional simplex given by x1 = t , /3, 5, + * * * +/3,,5,,5 cp(.r), xi 1 0 (i = 2 , . . ., n) is contained in S, and where cp(~,) = 4.
-+
Mordell dealt with the star body TI : Ix1Iu+* * lxnl" 5 1, 0 < CT < 1, and the star body T,: Clxitxil * xip(S 1, where the sum is extended over all sets of indices i,, . . .,$,with 1 S p S n , 1s i,< i,< ipsn.
-
*
*
a
<
364
CH. 5
SOME METHODS
34.4. Next, we give an application of theorem 3, with arbitrary p . Let fl, f , be convex distance functions in p and q = n - p variables, respectively, and let S, be the star body in R" determined by
(17)
( f i ( ~ i 7 *
-
-7
xp)}"
*
{fi(xp+
1
7 * * *7
Xn)}"
s1
or shortly f, 6 q ( f i ) , where q ( z ) = T - ~ / (T ~ > 0). Furthermore, let 5 be an arbitrary positive number. The tangent to the curve CT = q ( ~ ) at the point (t,cp(t))is given by CT
= p-v7,
where p = ( n / q ) t - p ' q ,
v = (p/q)5-"/q.
We wish to estimate A(S,) from below. A crude result, analogous to Minkowski's estimate for d(S,(@) (sec. 34.3), is obtained as follows. The star body S, contains the convex body K given by fz+vfl 6 p. By theorem 1, this body has volume V ( K ) = pv, vz/;v(p-vT)qTp-%
because p"v-" = n"(pPqq)-'. Observing that p = ( n / q ) ( - P / q p, / v = (n/p)r we may conclude (Groemer [34a]): for each 5 > 0 the star body S, and even the star body S ; ( ( ) given by
f19fi4 5 1,
fi
(n/P)57
fi
5 (n/q)c-P'q
has critical determinant 5 2-"n"p!q!(n!pPq4)-l Vl V , . We now improve upon this result by applying theorem 3. We suppose that p S q; this is no loss of generality. We take T~ = 2"lp5. Then we have
q(t) = and so q(5) 2 * p 2 cp(+zl).
P = (n/q)t-P'q, q(+zl) = 4t-p/4
Also, c p ( 4 = 3117
if we take z0 = ( 2 q / r 1 ) ~ /Next, ~ 5 . if $(z) is defined as in (Il), with the present choices of p, v, z0 and cp(z), then
A METHOD OF BLICHFELDT A N D MORDELL
0 34
365
Applying theorem 3 we get
Theorem 6. Let S2 be given by (17), with given convex distancefunctions fi,fz. Suppose that p 5 q = n - p . For 5 > 0, defiote by S 2 ( ( )the domain in R" given by
Then A(S2(5)) 2 V , V 2 u p , q ,where V i is the volume of the body fiS 1 (i = 1,2) and where up,qis the last member of (18). It can be shown that up,q> 2-"n"p!q!(n!pPqq)-'and that a,,,¶ Specializing theorem 6 to the case that fl(X1
,
* *
f2(xp+13
*,
x p ) = 1x11+ -
* * .,xn)
=
*
=
- + bPL
Ixp+lI+
* * .
+Ixnl,
one gets a result of Koksma and Meulenbeld [34b]. Some more special cases are dealt with by Mullender [34c]. 34.5. The foregoing theorems lead to interesting results in the theory of diophantine approximation, by the following reasoning. The number 5 appearing in the enunciations of theorems 4 and 6 is an arbitrary positive number. The lower bounds given for A(Sl(5)) and A(S2(5)) do not depend on this number 5. Letting 5 run through a suitable increasing sequence of positive numbers 5, we easily obtain the following
Corollary. Let /Ibe a positive number. Then each lattice with determinant
(
d(A) 5 1 + nlJ1(l+6),
+)
n+2
6 = (1-
,
366
SOME METHODS
CH.5
has infinitely many points in the domain
(20)
lxll - {max ( I X , ~ ,. . ., 1xnl))”-‘ 5 I, lxil 5 p
(i = 2 , .
. ., n).
Likewise, if p, q, fi,f 2 , V 1, V , and have the same meaning and satisfy the same conditions as before, then each lattice A with determinant d ( A ) < V 1V2Q ~ has , ~infinitely many points in the domain
(21)
(fl(x19
* -3
xp)}”
*
{.f2(xp+1,
. - * s Xn)}‘ 5
1,
f 2 ( ~ p + l ,
*, X n )
5 B.
The first assertion of the corollary also holds with the condition = 2, . . ., n) replaced by lxll 5 p. We observe that the inequalities (20) are a special case of the inequalities (21). Thus, apart from the numerical constants involved, the first assertion of the corollary is contained in the second assertion. The argument by which we proved the corollary can be looked upon from the point of view of the theory of automorphic star bodies (secs. 26 and 28). Actually, the star body S , defined by the inequality (17), with a given index p and given convex distance functions fi,f,, is fully automorphic in the sense of definition 26.3, with respect to the group of automorphisrns x’ = ax given by
lxil 5 fl (i
XI= o x i (i
=
1 , . . . , p ) , xi = o-PI4 xl (i = p + 1 , .
. ., n)
(w > 0).
It is of the finite type because, for some positive constant y, it is contained in the star body lxlxz * * xnJ5 y which is of the finite type. Moreover, S, is generated by the star body given by (21), for arbitrary p > 0. Therefore, the second assertion of the corollary is a consequence of the estimate A(&) 2 Vl V Z c c p , , and the first clause of theorem 28.7. In a similar way, the first assertion of the corollary follows. For further details we refer to sec. 45.2.
-
34.6. We recall that theorem 3 is a generalization of theorem 2 to the case of n-dimensional bodies. A generaIization of a different type was given by Mordell [34a] who considered star bodies S:
(22)
Ixnl
5 d l x l I J*
*
*r Ixn-1I)y
where q ( x l , . . ., xn- 1) is non-negative, convex and steadily decreasing in each of the variables, in the hyperoctant x i 5 0 (i = 1, . . ., n). For such star bodies he proved
§ 35
361
A THEOREM OF MACBEATH
Theorem 7. Let (tl, . . ., C;,) be a point on the boundary R of S, with t i > 0 (i = 1, . . ., n). Let Cy='=l xi/ui = 1 be the equation of a tac-plane to R at the point (tl,. . ., t;,,). Suppose that <,,/a,, 2 3. Then one has (23)
d(S)
2 alaZ
-
*
cr,/n!+2"1/,
where V is the volume of the domain
xi 2 ti ( i = 1 , . xn
5 r~(x1, *
n
. ., n),
C xi/cli 2 3, i= 1
xn-1)-3an-
The proof is very similar to that of theorem 2. Mullender [34b] constructed a star body T, with 9 T c S, which for a special class of domains S leads to sharper results. If S is the domain lxlxz * x,,l S 1, then this star body T is the union of all bodies T(i, , . . ., ip):
- -
{p-'(IxilI+
*
.
+IXi,I)+~P-ln)PIXi,+l
*
xi,!
5 1,
(xik!
21
( k = p + 1 , . . ., n),
wherep is any index with 1 5 p 5 n- 1 and (il,. . ., in)is a permutation of the set (1, . . ., n). Also, Mullender [34a] improved slightly upon theorem 2 and 3 by making a more favourable choice for the function +. 35. A theorem of Macbeath 35.1. In chapter 7 we shall deal with Minkowski's theorem on the product
of two iahomogeneous linear forms. A proof of this theorem by Macbeath [47a] consists in the construction of a sequence of points x' = (xlr, xzr), all belonging to a given grid r, such that (xlrxZr[ decreases rather rapidly. The construction is continued until one gets a point X with lxlrxZrl5 p, where p is a certain positive constant. Elaborating this idea Macbeath [35a] found a general theorem on grids which, in a number of cases, can be used to estimate from below the critical determinant or the inhomogeneous determinant of a given domain. This theorem reads as follows.
Theorem 1. Let r = z f A be a grid and let H be a convex body which is pos.sibly unbounded, but has n independent tac-planes. Suppose that M = H n r is not empty. Then there exists a point x E M such that the o-symmetric convex domain ( H - x) n (x - H ) has critical determinant
368
SOME METHODS
CH.5
5 d(I') = d ( A ) and therefore has volume (1)
V ( ( H - x ) n ( x - H ) ) g 2"d(A).
We reproduce the simple proof of this theorem given by Rogers [35a 1.
Proof of theorem 1. We call a point x E M an exterior point of M if there exists a closed half-space F bounded by a hyperplane P such that x E P, M n F = 1x1.
(2)
First we prove that M has exterior points. Let PI, . . ., P, be n independent tac-planes to H. It is no loss of generality to suppose that o is the common point of these tac-planes. Let G denote the convex cone containing H which is bounded by PI, . . ., P,, . Take a point z in the interior of G and a hyperplane P containing o such that P has only the point o in common with the cone G and does not contain a point # o of the lattice A. Each point of G belongs to some hyperplane Az P, A 2 0. Furthermore, each such hyperplane cuts off a bounded portion of G and does not contain two distinct points of r. Hence there exists a smallest value of A such that Az+P contains a point x of M = H n r. Clearly, this point x is an exterior point of M. Next, we prove that each exterior point of M satisfies the assertions of the theorem. Let x be such a point. Take F and P such that ( 2 ) holds. Then ( M - x ) n ( F - x ) = {o} and so o is the only point of I'-x = A in the set ( H - x ) n ( F - x ) . In other words, A is admissible for the set ( H - x ) n ( F - x ) . It is also admissible for ( x - H ) n ( x - F ) . Since ( F - x ) u ( x - F ) = R", it follows that A is admissible for ( H - x ) n n ( x - H ) . From this the assertions of the theorem follow.
+
Remark. If H is closed, then, clearly, the set ( H - x ) n ( x - H ) has critical determinant < d(A) and so (1) holds with the inequality sign. 35.2. Macbeath gave the following applications of this theorem.
I. H is the cube [ x i [ 5 I ( i = 1, . . ., n) and r = A is an arbitrary homogeneous lattice with determinant d(A) < 1. The intersection H n A is not empty, and ( H - x ) n ( x - H ) is a parallelepiped of volume 2"nY(1 - IxJ) (x E H ) . So we have the result that each lattice A with determinant d(A) < 1 contains a point x such that
5 35
369
A THEOREM OF MACBEATH
fi (1 - Ixil) < d(n)*
(3)
1
Clearly, x #
0,
because d(A) < 1.
11. H is the hyperoctant xi > 0 ( i = 1 , . . ., n ) and r is arbitrary. Now, for X E H, ( H - x ) n ( x - H ) is a parallelepiped of volume 2"x1x2 x,; theorem 1, together with the remark, leads to the following result of Chalk [50a].
-
Theorem 2. Each lattice and, more generally, each grid r contains a point x with x i > 0 ( i = 1, . . ., n), x1x2 * x,, 5 d ( r ) and also a point y with y i 2 0 ( i = 1 , . .)n), y 1y 2 * * * y,, < d ( r ) .
- -
.
111. H i s the convex domain (4)
xi > 0 (i = 1, .
. ., n),
xlxz
x,
* * *
2a
(a
> 0 constant).
We consider the set (H-x) n ( x - H ) , for a given point x E H. By the transformation y i = xiy; ( i = 1, . . . n; x = (xl . . .,x,,) fixed) it is transformed into the set (5)
--
where we have put 9 = a//?, /? = xlx2 x,, (so that 0 < 9 If q(9) denotes the volume of this set, then
5 1).
with p = x1x2 * * x, , 'V((H-x) n ( x - H ) ) = pq(a/j?), By studying the behaviour of q(9) for 9 -+ 1 one can prove that there exists a constant K, > 0 such that prp(a/p) 5 2" if a is sufficiently large and p = a(1 + (K,,a)-'/("+')). So one has /? 5 a(1 + ( i ~ ~ a ) - ~ / ( " + ~ ) ) if a is sufficiently large and V( (H-x) n (x-H)) 5 2". Applying theorem 1 and using homogeneity considerations one gets Theorem 3. An arbitrary grid r contains a point x with
d(r)
xlx2
--
X,
5 a(i+(~,a)-~/("+l))d(r),
provided a is positive and sufficiently large.
Macbeath also applied theorem 1 to the sets X1-(X2)Z-
(x1)2-(x2)2-
*
-
*
* -(XJ2
-
-(XJ2
2 0; 5 0.
370
CH. 5
SOME METHODS
35.3. Macbeath's original proof of theorem 1 is more complicated, but yields some extra information. Therefore, we reproduce it here. Let r, H satisfy the conditions of theorem 1. Consider the function 'p defined by (6)
cp(x) = V ( H , x ) = v ( ( H - ~ ) n (x-H))
(x E H ) .
It is non-negative on H and unbounded on each unbounded convex subset of H. By applying the theorem of Brunn-Minkowski to sets ( H - x ) n (x- H ) one finds that {'p(x)}"" is a concave function 0f.x and that this expression, when restricted to a closed bounded convex subset of H, attains its maximum at one point only. Now let xo be an arbitrary point in the interior of H. Let G(xo) denote the set of points X E H with p(x) 2 p(xo). Then G(xo) is a convex set, and xo lies on the boundary of this set. We take a tac-plane P to G(xo) at the point xo and denote by H' the half-space which is bounded by P and does not contain G(xo). Then the function cp is bounded on H n'H' and therefore H n H ' is bounded. So there is a tac-plane P' to H which is parallel to P and is contained in the interior of the half-space H'. Let it be given by P' = P - a . We now put
(7)
K = H n H', K' = 3 ( K + x 0 ) , K, = H n (H'+a).
Thus K is the part of H cut off by the hyperplane P , K, is the part cut off by P + a , and K' is obtained from K by a dilatation, with a factor +, with respect to the point xo. From the definition of K, K, and the function cp it is clear that (8)
q(H,x ) = q ( K 1 ,x)
if x E K .
We investigate, in particular, the quantities $ ( K , ) and z(K,) given by
(9)
$(Ki) =
XSK
q(Ki 3 X)Y 7(Ki) = max q(Ki Y x)* x E K\K'
We have z(K,) c $(K,) because the first maximum in (9) is attained at the point x = x o only. We now take into consideration the collection of all possible sets K , and use a compactness argument. The set Kl is bounded and convex. The hyperplanes P - a , P + a are tac-planes to Kl, and xo E P is an inner point of K,. The function cp(K,, x) takes its maximum on the set K at the point xo only. Considering parallelepipeds containing Kl we see that there exists an affine
0 35
371
A THEOREM OF MACBEATH
transformation A which sends x o into o and transforms K , into a convex set L , = A K , with the following properties 1". the hyperplanes x1 = f 1 are tac-planes to L , ; 2". the set L , is contained in the cube lxil S 1 (i = 1,. . ., n ) and has volume V ( L , ) 5 l/n!; 3". o is an inner point of L , and the function q ( L , , x ) attains its maximum on the set { x : x E L ,, x , S 0 } at the point o. The convex bodies L , possessing the properties 1 2", 3"form a compact metric space* in the metric 6 given by (25.3). The functionals $ and z defined on this space are continuous in the metric 6 and positive. Now the quotient T(K,)/$(K,) is left invariant under an affine transformation. Since this quotient is always less than 1, if follows that there exists a positive constant A,,< 1 only depending on n such that T ( K , )< l,,$(Kl). Thus we have O,
(10)
q(H,x) 5 1,,q(H, xo)
for x E K\K'.
This relation holds, with a fixed constant A,,< 1, for each point x o E H and a corresponding pair of sets K , K'. The proof of theorem 1 is now completed as follows. By the theorem of Minkowski-Hlawka (sec. 19). we have
(11) ~ ( ( H - x n ) ( x - H ) ) Iv ( ( H - n ~()x - H ) ) = q ( ~x), (x E H ) . We wish to show that A ( ( H - x ) n ( x - H ) ) 5 d ( T ) for some point x E M . Take a point x1 E M and suppose that A ( ( H - x ' ) n (x' - H ) ) > > d ( r ) .Then A has a point # o in H - x1 and so r has a point # x1 in H , because XIE r. We also have q ( H , x') > 0, by (1 1). Hence, x1 is an inner point of H . Taking xo = xl, in the definition of H ' , we have that r has a point # x1 in K = H n H'. Then there is a point of r in K\K'. Let xz be such a point. It is a point of M and, by (lo), it satisfies the relation
5
.')If it happens that A ( ( H - x 2 ) n ( x 2 - H ) ) > d(T), then we repeat the process. And so on. At each step we gain a factor 1,. Therefore, after a finite number of steps, we get a point x' E M with d((H- x') n (x' - H))I 5 d(T). This proves the theorem. V(H, x')
I n q ( ~ 7
As Rogers [35a] remarked, the considerations of Macbeath contain implicitly the following result.
*
See the considerations of sec. 25.3.
312
CH. 5
SOME METHODS
Theorem 4. There exists a positive constant 1, < 1 only depending on n which has the following property. Ifr, H satisfy the conditions of theorem 1 and i f q ( x ) is dejined by (6), then each point x E H n r with cp(x)
<1 '; inf ~ ( z ) zeHnr
is an exterior point of H n r. (Such points x exist if and only if the infimum is positive). Indeed, if x were not an exterior point of H n r and if H ' were defined as absve, then H n H' would contain at least one point x' # x of
r
with cp(x')
5
An(p(x) < inf cp(z). zeHnT
Rogers investigated also under what circumstances the set H n r has at most finitely many exterior points. Macbeath's method may be used to give an alternative proof of the theorem of Cebotarev (sec. 49.2).
36. Comparison of star bodies in spaces of unequal dimensions 36.1. Hermite was the first who derived an upper bound for the homo-
geneous minimum of an arbitrary positive definite quadratic form (sec. 5.1). As Mordell [36a] remarked, Hermite essentially proved a certain inequality connecting the absolute homogeneous minima of two positive quadratic forms in n and n- 1 variables, respectively. Later, the method was applied to other forms, in particular by Mordell [36a, 36b, 36c] and Oppenheim [36a, 42~1.In this section we prove a general theorem exhibiting the method; it was given in a less general form by Armitage [36a]. Let S be a star body with distance functionx In the following, we shall operate with the absolute minimum A(S) rather than the critical determinant A(S) and also consider the absolute minimum %(Sn P ) of the ( n - 1)-dimensional star body S n P, where P is some ( n - 1)dimensional linear subspace of R".For z # 0, we shall denote by P ( z ) the hyperplane which contains o and is orthogonal to z. Furthermore, for a given lattice A = AY, we shall consider the polar lattice A* = A*Y (see sec. 3.5). We recall that {%(s)]-"= A(S). In particular, 1(S) = 0 if S is of the
§ 36
COMPARISON OF STAR BODIES IN SPACES OF UNEQUAL DIMENSIONS
313
infinite type. We also have (1)
A(S) = sup Al(S,A){d(A)}"".
The supremum in the right hand member of (1) is attained for some lattice A , on account of theorem 26.6. Let .4 be an arbitrary lattice in R". Let b be a primitive point of the polar lattice A*. Then A has a basis {a', . . ., a"> such that (2)
a i * b = 0 (i = 1,. . ., n - l ) ,
a " . b = 1.
Thus A has n - 1 independent points in the hyperplane P(b). We put
L = A n P(b). (3) Then L is an ( ( n - 1)-dimensional) lattice in the (n- 1)-dimensional space P(b). As such, it has a determinant d ( L ) defined as the (n- 1)-dimensional volume of a cell of L. Taking a cell of A spanned by a cell of L and the point a" we see that the point a" has distance d(A)/d(L)to the hyperplane P(b). Hence, since b is orthogonal to P(b) and a" . b = 1, (4)
Ibl = d ( m @ ) .
We now prove (Lekkerkerker [36a])
Theorem 1. Let S be a star body of the finite type, with a group r of automorphisms, and let Po be a hyperplane through the point 0. Suppose that the following conditions hold 1". the set of points z # o with A(S n P ( z ) ) = 0 is not dense in R"; 2". for each hyperplane P containing 0, with 1 ( S n P ) > 0, there exists an automorphism Q = Sap E r with QP = Po; 3". if Sa E r, then O* E r. Then there is apoint bo on the boundary of S with bo (5)
Po; moreover,
{A(S)>.-z 5 {A(S n Po))"-*lbol.
Proof. Let f be the distance function associated with the star body S and let A be an arbitrary lattice. Let b be a primitive point of the polar lattice A*; write S(b) = S n P(b), L(b) = A n P(b). From the definition of the minima A,(&
(6)
A ) , Al(5'(b), L ( b ) ) it is clear that
w,4 5 Al(S(% W)).
374
SOME METHODS
CH. 5
We shall estimate Al(S(b), L(b)).Thereafter, by an appropriate choice of b, we shall obtain a good estimate for Al(S,A ) . If it happens that A(S(b)) = 0, then necessarily A1(S(b), L(b)) = 0 (see the relation (17.17)). Therefore, it is no loss of generality to suppose that A(S(b)) > 0. Then, by 2", there exists an automorphism 52~r such that BP(b) = P o . We have Po = P(b*) if we take b* = Q*b.
(7)
The point b* belongs to Q*A* = (an)*.Applying (4) to the point b* and the lattice !& and I observing that SaS(b) = S n P o , QL(b) = Q A n P , we find that
(8)
Al(S(b)J(b)) = = I , ( S n p0,QA n P,) 6 A(S n P , ) { ~ ( Q An Po)}"("-') = = A(S n P0){Jb*Jd(S2~)}'""-'' = A(S n Po){~b*~~(~>}'~~"~''.
From (7) it is clear that b* # 0. We prove that f ( b * ) > 0. Suppose that f ( b * ) = 0. Let z be a point # o with A(S n P ( z ) ) > 0. Then, by 2", there exists an automorphism 0,E r such that Q,P(z) = P o . We have b'f = d * z , with some ct # 0, because P ( z ) is orthogonal to z and Po is orthogonal to b*. By 3", C2; is an automorphism of S . Hence, f ( b * ) = lalf(z) and s o f ( z ) = 0. Applying 1" we see that the last relation holds for all z in some cone with vertex at 0. Thus S contains some infinite cone, which cmtradicts the assumption that S is of the finite type. This proves that f ( b * ) > 0. We now put bo = ab*, where CI > 0 is so chosen thatf(bO) = 1. Then
lb*I = f (b*) lbol =f(b*)lbol = f ( b ) l b O / ,
f (bO) because C2* is an automorphism of S. We substitute this in the last member of (8). At this stage, we make a definite choice for the point b. We recall that b was required to be a primitive point of A*. Accordingly, by the definition of A,(S, A*), we can choose b in such a way that f ( b ) < A,(S, A*)+&, where E is an arbitrarily chosen positive number. Using ( 6 ) and (8) we find that &(S, A ) < A(S n Po){(Al(S,n*)+~)Ib~ld(A))"("-') =
6 A(S n Po){~(S)(~(l"))""+~)"("-')(~bo~d(/l))''("-').
(j 36
COMPARISON OF STAR BODIES IN SPACES OF UNEQUAL DIMENSIONS
Since this is true for each conclude that
E
375
> 0 and since d(A*) = { d ( A ) } - ' , we may
Al(S,A ) 6 A(S n Po){A")~bo~(d(A))l-l'"}'/'"-''= =
A(S n Po){"S)[b0~}"'"~''{d(~)}''".
The lattice A being arbitrary, it follows that A(S) S A(S n PO){"S)lbo~}'l'"-''.
This proves (5) and completes the proof of the theorem.
36.2. We make a number of remarks. Remark 1. Similar considerations hold if instead of Po, there are finitely many hyperplanes P(b'), . . ., P(b'), where b', . . ., b' lie on the boundary of S, such that each hyperplane P through 0,with A(S n P ) > 0, can be transformed by a suitable automorphism QETinto one of the hyperplanes P(b'), . . ., P(b'). In this case, instead of ( 5 ) , one gets the estimate
(9)
{A(S)}"-z 5 max {{A(S n P(bk))}"-'Ibkl). k = 1,.
. ., r
Remark 2. For the applications it may be convenient to replace the requirements lo, 2" by the following two conditions (confer remark 1): 1'. for each point z # o with f ( z ) = 0 m e has A(S n P ( z ) ) = 0 . 2'. i f z is apoint withf( z ) = 1, then there is an automorphism transforming z into one of r givenpoints b', . . ., b' on the boundary of S. It is clear that, in the case r = 1, the conditions 1' and 2' imply the validity of 2". Furthermore, it follows from the proof of theorem 1 that 1" is not needed if I f , 2' and 3" hold. Remark 3. Suppose that, with the notations of remark 2, none of the hyperplanes P(b'), . . ., P(br) contains a given coordinate axis, i.e., that none of the points b', . . ., b' lies in a given coordinate plane x i = 0. For k = 1, . . ., r, denote by P k the i-th coordinate of bk and by the domain in the hyperplane xi = 0 which is the projection of S n P(bk) onto this plane. Then one has, for k = 1, . . ., r,
316
CH. 5
SOME METHODS
On account of these remarks we have the following variant of theorem 1.
Theorem 2. Let S be a star body of theJnite type, with distance function r of automorphisms. Suppose that the conditions l', 2', 3" and the hypothesis of remark 3 hold. Then, if the numbers p k and the projections are defined as above,
f and with a group
The method can also be applied if S does not admit a group of automorphisms fulfilling the conditions of theorem 1 or theorem 2, although the result is less useful. Consider the function g defined by
(11)
g(z) =
{A(s n P ( Z ) ) ) " - ' ~ Z ~
(z
+ o),
g(o) = 0.
If this function is continuous, then it is the distance function of some star body T. We prove that
{n(s)}ii-l 5 A(T).
(12)
Let A , b, L ( b ) have the same meaning as in the proof of theorem 1. Then
A(&
A)
5 A(s n P(b), W ) )5 6 A(S
n P(b)) {d(L(b))}"("-" = {g(b)d(A)}'""- '1.
The point b E A* can be chosen such that g ( b ) is not much larger than 1(T,A*)
s A(T)(d(A*)}"" = A(T){d(A)}-l?
So we have A(S, A ) 5 { A ( T ) } ' ~ ~ " ' ~ { d ( A ) } 'Since ~ " . A is arbitrary, the relation (12) follows.
36.3. We give some applications. First, we take as S the unit sphere K,. Then the condition 1" of theorem 1 is trivially true and the conditions 2", 3" hold if we take as r the group of all rotations about 0. So we have the following result (Mordell [36a], Oppenheim [36a])
(13)
A(K,)
5 {A(~"-~)}(n-1)'(n-2).
Next, we consider the star body So given by lxlxz * * * x,J 5 1 (Mordell [36b, 36~1).Let r be the group of all automorphisms xi = mixi
(i = I, . . ., n) (wi real, w1 w2
-
* *
on=
k 1).
3 36
COMPARISON OF STAR BODIES IN SPACES OF UNEQUAL DIMENSIONS
377
We prove that the condition 2' holds. Let z be a point # o with * * z,, = 0 ; without loss of generality we may suppose that z1= 0 and that z,, # 0. Then P(x) has equation z,x,+ * * * +z,,x,, = 0 and the projection of Son P ( z ) onto the plane x,, = 0 is given by
zlz2
(14)
lx1xz
--
*
x,-,(Cr,x,+
--
*
+Cr,,-'x,,-')~
s 1,
where cti = zi/z,, (i = 2, . . ., n- 1). If fi = Ict,l+ * * * +Ict,,-l( and > 0 is arbitrarily large, then the domain (14) contains the parallelepiped given by z
1x11
5 pP,
lxJ
5z
(i = 2 ,.
. ., n-1).
This parallelepiped has volume 2"-'P-'z. It follows that the domain (14) is of the infinite type. The same is true for So n P(z). Hence, condition 2' is fulfilled. . The conditions 1' and 3", with r = 1, are easily verified. We now take b' = (I, 1 , . . ., 1) and project the intersection Son P(b') onto the x,,-axis. Applying theorem 2 we get the following result.
Theorem 3. I f S , is given by lxlx2 * * x,,l 5 1 and i f S , , - , is the (n- 1)dimensional star body gioen by Ixl x2 * x,, - (xl + . * . + x,, - 5 1, then
n:='
Armitage [36a] gave a similar result for a star body given by an inequality of the type {Fi(x)}"'5 1; here the ni are positive integers with ni = II and Fi(x)is a sum of nisquares (xj)'. He deduced a more complicated inequality for the critical determinant of the domain (x')2{(x2)2+ . * . ( x $ } 5 1. Some further applications will be mentioned in secs. 41.4,42.3,42.4 and 45.4 (Mordell [41b, 42g], Oppenheim [42c], Mullender [45a], Davenport [45cI),
1
+
36.4. In the final part of this section we deal with an entirely different
method of comparing bodies in spaces of unequal dimensions. This method leads to upper bounds for the inhomogeneous minimum p(K, A ) of a given convex body K with respect to a given lattice A and has applications in the theory of diophantine approximations (secs. 45, 51). It goes back to Mordell [24a] and is often called the method of the additional variable.
378
cn. 5
SOME METHODS
The method is illustrated by the following theorem (Mahler [13a], Hlawka [13~1).
Theorem 4. Let K be a closed bouaded o-symmetric convex body in R". Put (16)
p = [(2/1,)"V-']+1,
where A, = l , ( K , Y ) and V = V(K).
Then for each point z E R" there exist a point u E Y and an integer k such that (17)
1 I;k 5 p - 1
and u + k z E ~ K , where p = 2(pV)r"".
Remark. By Minkowski's theorem, AlV = V(A,K) S 2". Hence (2/Al)"V-l 2 1, so thatp is an integer >= 2.
-
Proof of theorem 4. We havep > (2/A1)"V-', and so p < 2 (2/Al)-' = = A,. Let now pl be any positive number with p < pl < A, and let K' be the (a-t-1)-dimensional body consisting of the points x' = = (xi,. . ., x, xn+1 ) with (18) X+Xn+lZEP1K, lXn+ll 5 P . Here z is a fixed point in R" and x is the point ( x l , . . ., x,,) E R". The body K' is a skew cylinder with height 2p and with an axis passing through the origin and the point ( - z , , . . ., - z n , l), where z l , . . ., z,, are the coordinates of z. We apply Minkowski's theorem to the (n 1)-dimensional body K'. It is clear that K' is convex and o-symmetric. Furthermore,
+
V(K') = 2pV(p1K ) = 2pp; > 2pp"V = 2"+l. Hence, there exists a lattice point u' = (u,, such that, if u = ( u l , . . ., u,,),
. . ., u,, u,+
# (0,.
. ., 0,O)
I U , + ~ I 5 p-1. Since such a lattice point U' exists for all p1 > p and since Y is bounded U+Un+1ZEpiK,
lu,+lJ < p
and
SO
and closed, it follows that there also exists a lattice point u' = = (ul,. . ., u,, un+,) # o with
(19)
U+Un+lzEPK,
IUn+11
6 P-1.
Suppose that un+ = 0. Then u # o and u E pK. But this is impossible, because p < 1 , . So we have un+, # 0. For reasons of symmetry, we may suppose that un+ 0. This proves the theorem.
,=-
4 36
COMPARISON OF STAR BODIES IN SPACES OF UNEQUAL DIMENSIONS
379
Using theorem 4 we can easily deduce an upper bound for p ( K , Y ) and thus obtain a new proof of theorem 13.2 (Hlawka [ 13~1).The argument is as follows. Let K have distance functionfand let z o be an arbitrary point of R". We wish to estimate minf(zo-u). It is no loss of generality to suppose that this minimum is attained for u = 0. Put z = (2/p)z0, where p is given by (16). Take a point u and an integer k for which the relations (17) hold. Then we have f(+PZ)
5 f ( U + 3PZ) 5 f ( U + k z ) +f((+P - k ) z ) s
hence
f(4P) 5 3PP
=
P(PW/".
By the arbitrariness of z o , it follows that
P ( K Y)5
p(pV)-"".
This is slightly stronger than (13.7), because (pV)-"" <
+A,.
Schneider [13a] deduced theorem 13.5, in the special case that H I = H , and these bodies are o-symmetric and convex, by considering an (n+ 1)dimensional body of the form x + x , + ~ z E I X , H, + ~ ~~ , 5 2. A further application of the method is a proof by Mordell [51a] of (the first assertion of) theorem 51.6.
NOTE ON CHAPTER 5 As we indicated in the preface progress in the geometry of numbers was by no means uniform. In several areas modern as well as classical ones new methods were developed. But there were no significant contributions to the subject matter of Chapter 5, viz. classical methods for the determination or estimation of critical determinants. For this reason we do not insert a supplementary section on 'Methods'. For references to several new methods in this area we refer to sect. iii, vi, xi and xiv.
SOME METHODS
In this chapter we deal with general methods for evaluating or estimating, from below, the critical determinant of a given set in k". We give various examples and, in a number of cases, also determine the critical lattices. However, the systematic treatment of some important classes of star bodies will be postponed to the next chapter. In general, we can say that the critical determinant can be determined effectively only for a limited class of bodies; as to star bodies of dimension n > 3, as a rule, we can only derive more or less sharp estimates. The last two sections partly concern inhomogeneous minima.
29. The critical determinant of a two-dimensional star body. Methods of Mahler and Mordell 29.1. We know already a method by means of which we can find the critical determinant and the critical lattices of a bounded convex star body K in the plane; in fact, the lemmas 22.1 and 22.2 tell us that the critical lattices are determined by the inscribed affinely regular hexagons of minimum area. We now deal with a more general method which applies to arbitrary bounded star bodies in R2.This method, too, was developed by Mahler [29a, 29bl. It is mainly of theoretical interest. Let S be a given bounded star body in R2,with boundary R, and let A be a given lattice having two independent primitive points xl, x2 on R. The points xl, x 2 need not form a basis of A . But there is.a point y 1 such that {xl,yl} is a basis of A. Take such a point y l . Then { X I , y} is a basis of A if and only if y = + y l kxl, for some integer k and some choice of the sign. Moreover, if {xl, y } is a basis of A, then the point x 2 can be expressed as follows:
+
x2 = u l x 1 + u 2 y (u l, u2 integers with u2 # 0). (1) By choosing y suitably we can obtain that (2)
u2
> 0, 0
Ul
< u2.
9: 29
THE CRITICAL DETERMINANT OF A TWO-DIMENSIONAL STAR BODY
331
Clearly, y is determined uniquely by the requirement that {XI, y } is a basis of A and that the coefficients ul, u2 in (1) satisfy (2). It is not difficult to express the points of A in terms of XI,x2. Indeed, if y is determined as above, then the points of A are just given by the points x = ~ ~ x ' + t=~U ~; ~y ( ( ~ ~ U ~ - Z ~ +~ ZU ' ~~ )XX~ ~)
( Z ) ~ , I I integers). ~
Thus A is the set of all points x = u2- 1 ( W 1 X ' + W 2 X 2 ) ,
(3)
where w l , w2 are integers with w1 + w2 u1 = 0 (mod u2). We now state and p rwe a theorem which says that, for given x', x2 E R, there are only finitely many S-admissible lattices containing xl, x2 and that these lattices can be determined in a finite number of steps. The admissibility of A entails that XI,x2 are primitive points of A. In the following, fGr p > 0, we denote by C , the disc 1x1 5 p .
Theorem 1. Suppose that C, c S c C,,, where 0 < p < u, and let xl, x2 be two independent points on the boundary R of S. Then the S-admissible lattices containiflg XI,x2 are found among the lattices A having a basis {XI, y } such that the coeficients u l , u2 in the representation (1) satisfv (4)
On the other hand, such a lattice A is admissible, provided int S does not contain a point x # o of A such that (5)
lWil
6 a2{d(A)}-'
(i
=
1,2),
where w l , w2 are determined by (3). Proof. First let A be an S-admissible lattice containing x', xz and let y , u l , uZ be determined such that {XI, y } is a basis of A and ul, u2 satisfy (1) and (2). Then ldet (xl, xz)I = u,d(A) 2 u2d(C,) = ,/+
- u2p2.
We also have lxil 5 u (i = 1,2). Hence, u2 5 ,/$* ( ~ / p ) Thus ~ . (4) holds. Now suppose, in addition, that A does not contain a point x # o
332
CH.5
SOME METHODS
of int S such that Iwi( a'{d(A)}-l ( i = 1,2), where w l ,w2 are determined by (3). If x is a point of A such that IwlJ > a2{d(A)}-', then ldet (x, >I)'.
= u l ' . Iwl det (x*,I>)' .
= Iw,ld(A)
> a' 2 alx21.
and so 1x1 > a, so that x $ S . Likewise, x $ S if x e A and lw21 > > a2{d(A)}-'. It follows that A is admissible for S. This proves the theorem. Mahler considers in more detail the case that R consists of finitely many analytic arcs R 1 , . . ., R, separated by points b', . . ., bP on R. Thus, for i = 1, . . ., p , a variable point x E R i is an analytic vector function of a real parameter a. This parameter a can be chosen in such a way that the derivative x' = x(a)' does not vanish and so determines the tangent to R at the point x, for x E R\Z. Here, Z is the set {b', . . ., be}. Now let A be an extremal lattice for S. It contains two independent points x ' , x2 of R ; these points determine, on their turn, two integers u l , uz . The integers u l , u2 are bounded, by theorem 1. Mahler now proves that there is a finite set of pairs of equations with the following properties: 1". each pair xl, x 2 associated with an extremal lattice for S, with fixed values of u l , u 2 , satisfies one of these pairs of equations; 2". the pairs of equations can be written down explicitly; 3". each pair of equations is satisfied by at most finitely many pairs x', x2 and at most finitely many continuous sets of pairs xl, x 2 on each of which det ( x ' , x2) is constant. To give an idea of the equations occurring here we mention the following. If A has just two pairs of points +xl, f x 2 on R and these points do not belong to Z, then det ((XI)', x2) = det (xl, (x')') = 0 (this is the tac-line condition of lemma 27.1). If, however, A has just three pairs k x ' , f x 2 , +x3 on R and these points do not belong to Z, then one has an equation of the form x 3 = u ~ ' ( w , x ' + w 2 x 2 )E R and one further equation det (x2, x3) * det
(XI, (x')')
- det ((x3)',
+det ((x')', (x')')
(XI)')+
det
((XI)', x3)
det (x3, x') = 0.
29.2. Next, we consider asymmetric star bodies in the plane.
5 29
THE CRITICAL DETERMINANT OF A TWO-DIMENSIONAL STAR BODY
333
Let H be a bounded star body, which contains a neighbourhood of the origin and is not necessarily o-symmetric. Then S = H u (- H ) is symmetric with respect to 0,and also a star bJdy. Furthermore, S and H have the same set of admissible lattices. Hence,
d(H) = @),
(6)
whereas H and S have the same set of critical lattices. This reduces the present case to the previous one. Mahler [29c] deals with the particular case that H i s convex. This case gives rise to the following considerations. Let H be a closed bounded convex domain containing a neighbourhood of 0. Suppose that H is strictly convex and that the boundaries of H and - H have at most a finite number of points in common. Let R denote the boundary of S = H u (-If). Put
R+=RnH, R-=Rn(-H),
C=R,nR-.
The following theorem tells us how we can determine the critical determinant and the critical lattices of H.
Theorem 2. Let H.furfi11 the conditions stated and let A be a critical lattice. Then, i f A is singular, allpoints of A n R belong to C ; i f A is regular, then R , (and similarly R-) contains exactly three points x', x2, x 3 of A ; furthermore, for suitable choices of the signs, x3 = +x'+x2.
Proof. In virtue of our hypotheses, a tac-line to S = H u ( - H ) at a point x ECR \ has no point # x in common with S. Hence, by lemma 27.1, it does not belong to any singular critical lattice. Now let A be a regular critical lattice. Since R, and R- are the reflections of each other into the origin, it has at least three points x', x2,x3 on R+ . Furthermore, it follows from our hypotheses and from the theorem on lattice triangles (sec. 3.3) that each two points of A n R , constitute a basis of A. Therefore, x 3 has the form x3 = f x ' + x 2 ; moreover, R+ does not contain more than three points of A . This completes the proof of the theorem. A simple example is afforded by the circular disc H I defined by (7)
( x , -a)2
+(X,i2
5 p2,
One finds that A (H,) = 3(p2 - a')*
0 < CI < p.
+ +
(2a (a2 3p2)*). There is a unique
334
SOME METHODS
CH.
5
critical lattice; it contains the points of intersection of the boundaries of H I , - H I . Mahler [29d] studies in detail the star body S which is the union of two o-symmetric ellipses (8)
a i ( x , ) 2 + 2 P i x l ~ 2 + ~ i ( ~52 )12
( i = 1,2)
2 which both have area 71 (so that ct,y,-fl: = a,y,-Pz = 1). By applying a suitable linear transformation, of determinant 1, we can transform the two ellipses into the special ellipses E l , E , given by
(8')
(X1)Z+(X2)2
5 1,
B
- (x1)2+a--'
(x,),
r 1,
where a is some positive number. The transformation meant does not affect the simultaneous invariant (9)
J =
%YZ--2PlPz+azrl*
Therefore, we have J = a+o-'. The problem is to compute A(El u E,), as a function of J. In doing so we ma] suppose that the ellipses are not identical, so that B # 1 and J > 2. We may even suppose that B > 1 We prove
Theorem 3. The critical lattices of El u E, all belong to the set 9 oj lattices A possessing the following properties: 1". +J3 4 d ( A ) 5 1 2". A has bases {a, b ) , {c, d } with a, b E R 1 ,c, dE R,, where R idenotes the boundary of Ei (i = 1,2). Furthermore, each lattice in 9 is admissible for El v E,.
Proof. A lattice A which has determinant d(A) 2 3J3 and has two independent points on the boundary of El is generated by these two points and is admissible for El (confer the proof of lemma 22.1). A similar property holds for E, . Hence, each lattice A E Y is admissible for El u E,. Now let A be critical for El u E,. Then A is admissible for E l , and so d(A) 2 )J3. First suppose that A is singular. Then, by the argument used in the proof of theorem 2, its points on the boundary are found among the points of intersection of R , and R 2 . There are just four such points, and, by the theorem on lattice triangles, A is generated by these points.
9: 29
THE CRITICAL DETERMINANT OF A TWO-DIMENSIONAL STAR BODY
335
Next, suppose that A is regular. Then it has a t least 6 points on the boundary of El u E,. If there are 6 on the boundary R, of E, , then A is critical for El , and so d(A) = 3J3, so that A is critical for E,, too. If exactly 4 points of A lie on R1 and exactly 2 on R,,then, by a suitable small rotation (leaving invariant El), we get a lattice A' which is admissible for El v E, and which has just four points on R1 and no points on R,. By the foregoing, A' is not a singular lattice. Thus X is not critical, against hypothesis. Thus, in all cases, at least 4 points of A belong to R,. By the theorem on lattice triangles, two independent points among them constitute a basis of A . The same property holds for the points of A on R,. It is clear that a lattice A possessing property 2" has determinant d(A) 5 1. The proof of the theorem is now complete. For the points a, b, c, d appearing in 2' we have relations of the type c =ula+uzb,
d =ula+u,b
( u 1 , u ~ , u 1 , uintegers). z
Mahler derives estimates for IuJ, luil ( i = 1,2) and determines the set of lattices A satisfying 1" and 2", for 2 < J < 25. Thus A(E, u E,) is known for 2 < J < 25. For J + co,A(E, v E,) tends to J3.
+
29.3. Mordell [29a, 29b] has developed a method which applies to not too restricted a class of two-dimensional star bodies S a n d which, in many cases, leads to the exact value of A(S). The method is not restricted to bounded star bodies and can be described as follows. Let S be a two-dimensional star body, with distance function f and with a boundary R consisting of four arcs R,,. . ., R4,such that the following requirements are fulfilled: 1". S is symmetric with respect to the coordinate axes and the bisectrices
thereoj: 2". R,lies in the i-th quadrant and terminates on the coordinate axes or has the axes as asymptotes, for i = 1, . . ., 4. 3". Ri is a conuex curve,for i = 1 , . . ., 4. Some consequences of these requirements are as follows. The distance function f satisfies
336
CH.
SOME METHODS
5
on account of 1". If x = (xl, x 2 ) is a point of R , , then (.I
9
x2+4
4s
(x,+u, x2-c) E S
for u
=- 0,
if x1
2 x2 2 u > 0.
This implies that f(1,u) is steadily increasing for u > 0 and that f (1 u, 1- u ) is non-increasing for 0 5 u 5 1. Consequently,the function g given by g(u) = f ( 1 u, 1 -). - f ( k 0)
+
+
is steadily decreasing for 0 5 u 5 1. We also have
do) = f ( L 1 ) - - f ( 1 ,0) > 0 s(3) =f(% 4) > 0 4)-f(19
g ( 1 ) = f ( 2 , O )- f ( L 1 ) Hence, there is a real number 9 with 3 are two positive numbers a, j? with
(11)
a2+B2 = I,
$01
s 0.
-= 9 5 1, g ( 9 ) = 0. Hence, there
< /I
a,
g(/I/a) = 0.
The last relation (1 1) is equivalent with For p > 0, the star body p S also satisfies the requirements lo, 2", 3". It has distance function pf, but leads to the same pair of numbers E, p. Therefore, it is no loss of generality to suppose that
(13)
f ( a , B ) = 1.
Now consider the 16 points fa', fb', given by
(14)
a' = (a-B, a+B), a3 = (a, B),
a' = (a+/?, -(a-B)), a4 = (B, -a),
b' = (a+B, a-B), b3 = (B, a),
b2 = (-(a-B), a+B), b4 = ( - a , B).
By (12) and (13), all these points belong to R.The points +a', +a2 and the points f a 3 , +a4 are the vertices and the mid-points, respectively, of the sides of a certain parallelogram P,,and they generate a certain lattice A , . Similarly, the points fb' determine a parallelogram Pz and
9: 29
THE CRITICAL DETERMINANT OF A TWO-DIMENSIONAL STAR BODY
337
a littice A , . These lattices have determinant 1, and the parallelograms P , , P , have area 4. Resuming we may say that the requirements 1"-3" lead to the existence of two configurations of 8 points all lying on the boundary, such that the points of each configuration are the vertices and the mid-points of the sides of a certain parallelogram Piand so determine a certain lattice A , ( i = 1,2); if the normalization condition (13) holds, then these lattices have determinant 1. For a large class of domains S, the lattices A , , A , are admissible. If it happens that there are no S-admissible lattices of smaller determinant, then A , , A , are critical lattices and so d(S) is known. Mordell expresses the last condition in a manageable form. In this way he obtains the following Theorem 4. Let S be a star body satisfying the requirements lo,2", 3". Let a, p > 0 be determined by (12) and suppose that (13) holds. Denote by RT the sub-arc of R, with end-points a3 = (a, p), b3 = (p, E ) . Then the following statements are true. a) For each point x E RT , there are two points y , z on R,, such that y - z = x. b) If the parallelogram oxyz has area 2 1, for each point x E R:, then A ( S ) 2 I ;if in addition, the lattices A , ,A , introducedabove are admissible for S, then A ( S ) = 1 and A , , A , are critical lattices. We do not give the proof of this theorem which is of an elementary character, but rather lengthy. The main idea of the proof is to apply Minkowski's theorem to an arbitrary S-admissible lattice of determinant 1 and each of the parallelograms Pi considered above and to consider suitable linear combinations of the lattice points thus obtained. The applicability of the theorem is, of course, restricted by the fact that we are looking for critical lattices which have a given configuration of points on the bmndary of S. In some instances it may be advantageous to work with different configurations (see the paper by Clarke cited below). The method may be applied to the star body Ix1x21S 1 (which will be handled, in a different way, in sec. 41). Mordell applies his method to several other domains. We state some of his results. (See also Cassels
[GNI.)
338
CH. 5
SOME METHODS
I. Circular quadrilateral S1, with distance function fl(X1
9
x2)
= I ~ l I + l ~ s l + { ( P 2 - 1 ) ( ( ~ 1 ) 2 + ( ~ 2 ) 2 ) + 2 ~ 1 ~ 2 }(1 *
Critical determinant A($,) critical lattices.
=
s P
{p2+2+2(--p4+ I2p2-4)+)-'.
Just two
11. Star-shaped octagon S, , with distance function fi(X1, x2) =
min (IX,l+PlX,l,
(3-*
IX21+PIXlI)
5P
s 1).
Critical determinant d(S,) = (5-4p+p2)(1 + 2 p - p 2 ) - , . Just two critical lattices if 3-* < p 5 1; four if p = 3-*. Clarke [29a] extends Mordell's method and determines d(S,) for 2-3i I_ p I_ 3*. Similar methods, for other classes of domains, were developed by Bambah [29a] who investigated star bodies in R2 with hexagonal symmetry and by Mordell[29c] who treated star bodies resembling the domain l(x1)3+(x2)31 5 1. 36). Some special two-dimensional domains
30.1. We list several separate results. I. The union S,(a, B ) of the four circular discs (x1)2+(x,)2+ctxl
5 0,
(XJ2+(XZ)
2
+Px2
50
(a
> P > 0).
Critical determinant d(Sl(a, p)) = ct3/3(a2+ P 2 ) - l . Just two critical lattices, each of which is generated by the point (a, 0) and an end-point of one of the four circular arcs which make up the boundary of ,",(a, /I). (Ollerenshaw [30a].)
11. The quadrifoil S2(a) which is the union of the four discs (Xl+a)Z+(x2)2
5 1,
(X1)2+(X,fa)2
61
(0 < ct 6 1).
Here, for 5-* 5 ct 5 1, there is just one critical lattice, generated by the two pairs of points separating the four circular arcs of the boundary. For 0 < a < 5-*, there are two regular critical lattices each containing one of the pairs of points just meant. (Ap Simon [30a].)
I 30
339
SOME SPECIAL TWO-DIMENSIONAL DOMAINS
111. The union S3(a)of the three domains
(x1)z+(x2)z 6 az,
(X1)z+(Xz)zfX1
50
(0
< a < 1).
One has d ( S 3 ( a ) )= max { a ( l -az)*, 2a3(l - az)*, 3a2J3}. (Ollerenshaw [30bi.)
IV. ThestarbodyS,(a)={x: Jxlxz151, l ( ~ ~ ) ~ - ( S x 2~a)}~(O
+
V. The star body S,(a) = {x: Ix1x21 S 1, (x1)z+(xz)2S a } ( a > 0). One has d(S,(a)) = +aJ3, (2a- I)*, 2, (2a-4)*, J5, successively in the a-intervals 0 < a S 2, 2 6 a S 3, 3 S a 5 4, 4 S a =< 9, c1 2 4. (Ollerenshaw [30d].) VI. The star body &(a) = {x: - 1 S x1xz 5 a, Ixl + xzl 5 (a’+ ) ’ . 4 ( a > 0). Critical determinant d(S,(a)) = ( a ’ + 4 ~ ~ ) Two ~ . continuous sets of critical lattices; each lattice from any of these sets has a cell entirely contained in &(a). (Cassels [30a]; confer secs. 27.4 and 45.6).
VII. The star body S7(a) = {x: lxll S 1, 1x21 S l + a x l * (1-21x11)}
(0 < c1 5 3). Critical determinant d(S7(a)) = 1. Same remark as in the previous case. (Cassels [30a].)
30.2. There are also results for domains which are not star bodies.
V I E The domain &(a) = {x: Ixlxzl 1, xz 2 a> ( a 2 0). Critical determinant d(S,(a)) = J 5 , for all a 1 0. The union of S8(a) and the strip 0 S x1 5 a has critical determinant 3(3 + J 5 ) , for all a > 0.
The first result follows from the theory of sec. 28; the second assertion is based on a result of Prasad [45a] concerning the approximation of irrational numbers by rationals. (Mahler [30a].)
IX. The square frame & ( a )
= (x: a
5 max(Ixll, Ixzl) S I} (O<
a< 1).
340
SOME METHODS
CH.
Critical determinant d(S,(ci)) = m-’, where rn = [(1- a ) - ‘ ] . 4 S ci c 1, there is only one critical lattice. (Ollerenshaw [30e].)
5
For
31. The critical determinant of an n-dimensional domain
31.1. In this section we develop, first of all, Minkowski’s method of evaluating the critical determinant of a convex body in R 3 (Minkowski [31a]). We begin by studying lattice octahedra. Let A be a lattice in R 3 and let P be an octahedron with centre a t 0. If the 6 vertices of P are points of A , but no other points # o of A belong to the interior or the boundary of P, then P is called a lattice octahedron (with respect to the lattice A ) . The following theorem says that there are two different types of lattice octahedra. Theorem 1. Let P be a lattice octahedron, with respect to a given lattice A , andlet fx i ( i = 1 , 2 , 3 ) be the vertices ofP. Then A has a basis {a’, a’, a 3 ) , such that one of the following two possibilities arises: I. x i = a ‘ f o r i = I, 2 , 3 11. x i = a’for i = 1 , 2 a n d x 3 = a1+a2+2a3. We shall say that P is of the type I or 11, according to whether case I or case I1 occurs. In the last case, a‘ = x l , a2 = x 2 , a3 = +(- x l - x 2 + x 3 ) , so that A consists of all points x of the form with u 1 = u2
x =~(U1X1+UzX2+U3X3),
(1)
= u3 (mod 2).
Proof of theorem 1. By the corollary to theorem 3.3, A has a basis (a‘, a’, a 3 ) ,such that the three vertices x l , x z , x 3 of P have the following form: x1 = U
l d ,
x2 = v l a1 +uza 2, x3 = w 1 a 1 + w 2 a 2 + w 3 a 3 ,
where u1
> 0, 0 5
Vl
c
212,
0
5
w1
< w 3 , 0 2 w2 c w 3 .
Now, since P is a lattice octahedron, the triangle ox1x2does not contain other points of A. Hence, by theorem 3.4, u1 = v 2 = 1, and so o1 = 0. Therefore, it is permitted to interchange al, a’; so we may suppose that w1 w2, It is convenient to write now w1 = p , w2 = q, w3 = r. We apply the linear transformation B determined by Ba’ = e’ (i = 1 , 2 , 3). We have
s
§ 31
THE CRITICAL DETERMINANT OF AN n-DIMENSIONAL DOMAIN
341
BA = Y, while BP is the octahedron with vertices f(1, 0, 0), f(0, 1, 0), f( p , q, r ) , so that BP consists of the points x satisfying (2) Here, p , q, r are integers with 0 S p S q < r, whereas ( 2 ) is not fulfilled for any lattice point u # f(1,0,0), f(0, 1,0), f.( p , q, r ) . We wish to prove that ( p , q, r ) = (0, 0, 1) or (1, 1,2). To this end, we distinguish three cases. (a) r = 1. Then, necessarily, p = q = 0. (b) r arz odd integer > 1. Then (2) is satisfied by x = ( k , , k, , l), where k, = 0 or 1, according to whether p < +r or p > +r, k, = 0 or 1, according to whether q < +r or q > +r. (c) r an even integer > 1. Put r = 2s. If not p = q = s, then (2) holds for some lattice point x = (k,,k , , l), k, = 0 or 1, k, = 0 or 1. If p = q = s a n d s > 1, then (2) holds f o r x = (1, 1,2). Onlyifp = q = s = 1, there is no new integral solution to (2). Summarizing the results we find that BP, and therefore P, has the required form.
31.2. Now let K be a bounded o-symmetric convex body in R 3 , with distance function$ Let A be a K-critical lattice. Then, by theorem 26.1, there are three independent pairs of points of A on the boundary of K. They determine an octahedron P. It may be that P is not a lattice octahedron. But P certainly contains such an octahedron, and the vertices of this new octahedron also lie on the boundary of K, because K is convex and A is admissible for K. We may conclude that, for each K-critical lattice, there exists a lattice octahedron, with vertices on the boundary of K. We deduce a converse to this assertion, under the condition that a certain finite set of points of A do not lie inside K. Theorem 2. Let P be a lattice octahedron, with respect to a given lattice A , and with vertices x i (i = 1,2,3) on the boundary of K. Then A is admissiblefor K if either 1". PisofthetypeIandf(x) 2 1f o r a l l p o i n t s x = u , x ' + u , x 2 + u , x 3 ~ A ,
342
SOME METHODS
+
CH.
5
+
where one of the ui is 0, k 1, 2 and the other ones are equal to 1, or 2". P i s o f t h e t y p e I I a n d f ( x ) 2 1forallpointsx = + ( + x ' + x 2 f x 3 ) . Proof. First suppose that 1" holds. It is no loss of generality to take x i = e' ( i = 1,2, 3). Then A = Y and f ( e i ) = 1 (i = 1, 2, 3); furthermore, f ( u ) 2 1 for the points u = ( u l , u 2 , u3) specified under 1". Assume that there exists a lattice point u # o with f ( u ) < 1 . For reasons of symmetry, we may suppose that 0 5 ui 5 u2 5 u3;by our assumptions, we then have u3 2 2 and u # (I, 1,2). It follows now from the proof of theorem 1 (see cases (b) and (c)), that the octahedron P,,with vertices + e l , k e 2 , k u contains some lattice point 0 = (ul, u 2 , 1) # + e l , f e z , +u, with v i = 0 or 1 ( i = 1, 2). Since f is convex and since f ( e l ) = f ( e 2 ) = 1 and f ( u ) < 1, we havef(0) < 1. This contradicts our assumptions. Thus f ( u ) 2 1 for all points u # o of Y , i.e., the lattice A = Y is admissible for K . Next, suppose that 2" holds. Again take x i = e' ( i = 1 , 2 , 3 ) . Then A is the lattice of points +u = +(u,, u 2 , u3) with u1 = u2 = u3 (mod 2) (so that A # Y ) ;further, f ( e i ) = 1 (i = 1,2,3) and f ( x ) 2 1 for the 8 points (++, +$, A+). From the relations f(O,O, 1) = 1, f(1, 1, 1) 2 2 it follows that f(1, 1 , O ) 2 1 , f(1, 1,2) 2 1. The last two relations remain true if the coordinates are permuted or one or more signs are changed. Thusf fulfills the assumptions of the previous case (at the points of Y , not at those of A ) . Therefore, f ( u ) 2 1 for all points u # o of Y. Now assume that f ( u l ++, u2++, u 3 + + )< 1 for some triple of integers u l , u 2 , u 3 . We may suppose that 0 5 u1 5 u2 5 u3 and that u3 2 1 . Then we have f ( $ , +,4) < 1, because (+,+,3) is a point of the octahedron with vertices + e l , + e 2 , k ( u , + $ , u2+$, u3++), contrary to our hypotheses. This contradiction proves that f ( x ) 2 1 for all points x # o of A, and so completes the proof of the theorem. The next theorem gives the possible configurations of the points of a critical lattice on the boundary of K .
Theorem 3. There exists a K-critical lattice A with a basis ( x , y , z ] , such that one of the followiizg two possibilities occurs:
I) the 12 points + x , + y , +z, + ( x - y ) , + ( y - z ) ,
k ( z - x ) lie on theboundary ofKandthepoints i . ( - x + y + z ) , + ( x - y + z ) , + ( x + y - z ) lie outside K ,
9 31
THE CRITICAL DETERMINANT OF A N n-DIMENSIONAL DOMAIN
343
11) the 12 points f x , + y , fz, f( x + y ) , f( y + z ) , f( z + x ) lie on the boundary of K and t ( x + y + z ) lie on the boundary of K or outside K. Moreover, if K is strictly convex, then each K-critical lattice possesses this propertv.
Proof. We consider an arbitrary K-critical lattice A , and distinguish two cases. By R we denote the boundary of K. Case 1. Each triple of independent points of A n R determines a lattice octahedron of the type I. Let {XI, x2, x 3 } be such a triple, and let Tl(A) denote the set of the 26 points x = u1x1 u2x2 u3x 3 , where
+
u i = 0 or f 1
+
( i = 1,2,3)
and (ul, u 2 , u3) # (0,0,O).
We observe that the points +2x' f x 2 f x 3 do not belong to R (and therefore lie outside K ) , because the octahedron determined by such a point and the points x 2 ,x 3 is of the type 11. Similarly, the points + x l 5 2x2 fx 3 and the points + x ' + x 2 + 2 x 3 do not belong to R. Therefore, by theorem 2, the lattice A remains K-admissible, if it is subjected to sufficiently small variations such that the points x i remain on the boundary and the set T l ( A )remains admissible for K.This enables us to prove the following assertion (a) i f x , y , z are three points of A n R, then it is not true that eachpoint of T l ( A )n R is of the form
(3)
+z
or
+(x+v,z)
or
t-(y+u,z)
(01,02
integers).
Indeed, suppose that each point of T , ( A ) n R is one of the points (3). We consider two types of variations of A : a) x, z invariant; y varies on R in such a way that y+z, y - z do not become inner points of K,
p ) y , z invariant; x varies on R in such a way that x + z , x - z do not become inner points of K. Under such variations of A , all points of the form (3) remain outside the interior of K, on account of the convexity of K. Now it is always possible to perform an arbitrarily small variation of the type a ) or the type or, if necessary, a combination of such variations, such that A is transformed-into a lattice A' of smaller determinant. But, by our
a)
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CH.
5
assumption on Tl(A), the set T,(A) remains admissible under such variations. The same is true for A , if the variations are sufficiently small. This contradicts the hypothesis that A is a critical lattice and therefore proves the assertion. From the assertion (a) and the form of T l ( A ) it easily follows that there exists a triple of points x,y , x E T l ( A ) , such that x,y , z,x-y, y - z , z-x all belong to Tl(A) n R. None of the points -x+y+z, x-y+z, x+y-z belongs to R, because each of these points, together with two of the points x, y , z, x-y, y -z,z-x,leads to a lattice octahedron of the type 11. This proves that, in case 1, always the possibility I) occurs.
Case 2. There are three independent points in A n R determining a lattice octahedron of the type 11. Let xl,x2,x3E A n R have this property and let T,(A) denote the set of points f+xl& +x2f+x3.By theorem 2, the lattice A remains admissible for K, if we vary xl, x2,x3 on R in such a way that no point of T2(A) enters into the interior of K. Such a variation can not lead to a lattice of smaller determinant, because we start with a K-critical lattice. We consider three different variations. a ) Suppose that T2(A)n R is empty. We keep xl,x2 fixed and vary x3 on R in such a way that d(A) does not increase, until some point of T2(A)appears on R. Without loss of generality we may suppose that f(+XI +tx2++x3) become points of R.
p) Put x = xl,y = +x1++x2++x3,z = -x2 and suppose that the points of T 2 ( A )n R are all of the form fz, +x3 or klx+kzy(kl, k2 integral). We keep x,y fixed and vary the pair z,x3 = z+ (2y-x) on R in such a way that d ( A ) does not increase, until some new point of T z ( A ) appears on R.After a suitable permutation of the xi,we get the situation that the points f(f+x' ++x2++x3) all lie on R. y ) Define x, y , z as above and suppose that T2(A)n R contains the points + y , j-(y-x), but no other points. By a similar transformation as under 8) we obtain a situation in which T2(A)n R contains still more points, e.g. f(+x' -+x2+3x3). The remaining pair, viz. f(3x' t+x2-+x3),may lie on the boundary of K or outside K.
It follows from these considerations that, if necessary, we can transform A into another K-critical lattice (again denoted by A ) with the property that Tz(A)n R contains the points
9: 31
THE CRITICAL DETERMINANT OF A N n-DIMENSIONAL DOMAIN
345
whereas k 4('. +(3s2-+x3) lie on the boundary of K or outside K. If K is strictly convex, then our considerations tell us that the original lattice has already this property. The assertions of the theorem now follow on taking x = -1x1 + + x 2 + + x 3 , y = +x1-12x 2 + z1x 3, z = -*x1-+xZ-J$. 2 Remark. We observe that only in case 2 it is necessary to pick out a particular critical lattice. Presumably, the assertion of theorem 3 is generally true for all K-critical lattices, but this has never been proved. 31.3. The foregoing results can be used to evaluate, at least theoretically, the critical determinant of a given convex body in R 3 . Let K be a bounded o-symmetric convex body in R 3 , with boundary R. By the foregoing, there exists a K-critical lattice A , with a basis {x, y, z}, such that one of the following two possibilities arises: 1) each triple of independent points of A n R determines a lattice octahedron of the type I and, moreover, property I of theorem 3 holds, 2) property I1 of theorem 3 holds. We denote by Z the set of points of A n R enumerated in theorem 3 under I and 11, respectively. Thus Z consists of k pairs of points, where k = 6 in case 1) and k = 6 or 7 in case 2). Furthermore, as follows from the proof of theorem 3, there are three independent points xl, x 2 , x 3 E Z such that, in case I), each point x having the form * k x 1 k x 2 + x 3 or fx'+kx2+x3 or + x ' - t x 2 + _ k x 3 (k = 0 , l or 2) and not belonging to Z, lies outside K, whereas, in case 2), each point x $ Z of the form t- +xlk+x2 f x 3 lies outside K. Now let f x l , . . ., f x k be the complete set of points in Z and let H I , . . ., Hk be half-spaces not meeting the interior of K, such that xi lies on the boundary of H i , for i = 1, . . .,k. The fact that the points +XI, . . ., * x k lie on R yields k equations for the 9 coordinates of the points xl,x2, x 3 . To get more equations we consider small variations of A which are such that x i remains a point of Hi, for i = 1, . . ., k. By theorem 2 and the foregoing remarks on the set 1,the lattice A remains admissible for K, and so d(A)does not decrease, if the variation performed is sufficiently small. Thus ldet (xl,x2, x3)/ does not decrease if xl,x 2 , x3 are subjected to small variations not affecting certain k linear inequalities. This leads to 9 -k further equations for the coordinates of
*
346
cn. 5
SOME METHODS
xl, x2, x3. So, in the aggregate, there are 9 equations for these coordinates. Among the solutions of these equations there is a set {xl, x2, x3} de-
termining a K-critical lattice.
Minkowski sets up and solves the equations meant in the case of the octahedron Po given by (4)
Ix,l+lx2l+lx3l
5 1.
He finds that there are 8 critical lattices. They are transformed into each other by reflections into coordinate planes; one of them is the lattice generated by the three points al
=(-”6 ,
1 2) 62 6
u2
=(? 67
-2
1)
69 6 7
u3
= (L 6, 3 69 2 6 ) 9
so that (5)
d(P,)
=
det (u’, u 2 , u3) = -&$.
Accordingly, the density of densest lattice packing (sec. 20) of Po is given by
(6)
b(P0) = V(P,)/d(2P,)
= +$.
Whitworth [31a, 31b] dealt with the domains
lxil 5 1 (i = 1,2,3),
Ix1+x2+x3l
5 z (0 < z < 3);
5
{ ( X I ) ~ + ( X ~ ) ~ } ~ + I X ~ 1. I
The first domain is affinely equivalent to Po if z = 1. For both domains the numerical details of the procedure are rather involved. Minkowski [31a] also stated that the density of densest lattice packing of an arbitrary tetrahedron P I is equal to +3(Po) = 2K. This statement was shown to be wrong by Groemer [31a] who found that 8 ( P l ) 2 > 4%. Some authors have tried to generalize the foregoing theory to the four-dimensional case. Thus Wolff [3la] continuing work by Brunngraber derived four-dimensional analogues of theorems 1 and 2 and used the results to obtain the critical lattices and the critical determinant of the unit sphere in R4.Mordell [31a] gave a simple proof of the four-dimensional analogue of theorem 1. 31.4. The theory of sec. 29.1 can, to a certain extent, be generalized for n-dimensional domains. In particular, we consider convex bodies and
ri 31
THE CRITICAL DETERMINANT OF A N n-DIMENSIONAL DOMAIN
341
derive two finiteness theorems due to Cohn [31a]. For p > 0, we denote by C,, the sphere 1x1 S p . The volume of the unit sphere in R" is denoted by IC, (m 2 1).
Theorem 4. Let p be a positive number and let K be a (closed) bounded o-symmetric convex body in R" containing C,. Suppose that K does not contain a lattice point u # o with lul 5 cc,p'-", where u, = n 2 " - ' ~ ~ - ' ~ . Then Y is admissible for K (so that A(C,,) = p"A(C,) 1). Proof. Suppose that K contains a point x with 1x1 = a,pl-". Denote by H, the,(n- 1)-dimensional linear subspace of R" which is orthogonal to the vector x , and by K, the convex hull of f x and the (n- 1)-dimensional sphere C , n H,. The set K, is an o-symmetric convex body contained in K. It has volume
So it contains a lattice point u # 0 . This contradicts the hypotheses of the theorem. Hence, Kdoes not contain a point 1x1 with 1x1 = ~ l , , p ~ and - " so K i s contained in the sphere 1x1 < q,pl-". Applying again the hypotheses of the theorem we may conclude that Y is admissible for K. Theorem 4 can be applied as follows. Let K be a bounded o-symmetric convex body in R" and let A be a lattice having n independent points x l , . . ., x" on the boundary of K. By the corollary to theorem 3.3, there exists a basis {a1,. . ., a"} of A such that, for i = 1 , . . ., n, x i has the form xi = u l i a ' + . +uiiai, where uli,. . ., uii satisfy (7)
05
uki
< uii
(k = 1 , .
.y
i-1).
We consider the linear transformation A given by Ae' = a' (i = 1, . . ., n ) and put K' = A-'K. Then K = AK' and x i = Au', where (8)
i
u = (Uli,.
. ., u i i , 0,.. .)0)
( i = 1,.
. .)n).
The points d,. . ., u" all lie on the boundary of K'. Hence, K' contains the generalized octahedron P with vertices f ul, . . ., ku". Now suppose that A is admissible for K. Then X I , . . ., x" are a set of minimum vectors with respect to K . By (9.13) the index ldet (uki)l of this
348
CH. 5
SOME METHODS
set of vectors in A satisfies the estimate
-
ldet (uki)l = u l l u 2 2 * * u,, S n ! . (9) Clearly, there are only finitely many generalized octahedra P, with vertices fd,. . ., fu", such that the points uihave the form (8) and have coordinates u k i satisfying (7) and (9). The intersection of these octahedra contains a sphere C, , where p = pn is a positive constant only depending on n. Applying theorem 4 we obtain Theorem 5. The critical lattices of a bounded o-symmetric conuex body K, with distancefunctionf, are found among the lattices A having n indepen+ u i i a i on the boundary of K, where dent points xi = uliai+ {a', . . ., a"} is a basis of A and where the coeficients uki satisfy (7), ( 9 ) and
-
f(u,al+
(10)
for all u
= (ul,
. . ., u,)
. - .+u,an) 2 1
with 101 5 ct,,p;-"
=
n2"-i~n-lp~-".
By considering special lattices (admissible for the unbounded star body Ixll{max (IxzI,. . ., I x , , ~ ) ) "5- ~l), Cohn shows that theorem 4 is no longer true if a, is replaced by a sufficiently small positive constant & or ct,,p1-" by &p-", where p, > 0 and rs
c = (x = (x1 a x2 > xg): (x,, x2) E K , 1x31 5 11. (1) It is trivial that d(C) S d ( K ) . The question arises whether or not in this relation the equality sign holds. In sec. 20 we proved already theorem 20.6 which we restate as Theorem 1. If K is a bounded o-symmetric convex domain in R2 and C is deJined by (l), then d(C) = d ( K ) .
We mention some other proofs of theorem 1. We denote the boundary of C by R and the part of R in the plane x , = 1 by R,.
9: 32
349
SOME SPECIAL DOMAINS
Yeh [32a] deduced theorem 1 by using Minkowski’s theorem 31.3. Chalk and Rogers [32a] deduced it from the following two propositions: (a) There exists a C-critical lattice A , which has three points a, b, c on R, such that, in the plane X, = 1, c is an innerpoint of R , . (b) Let a‘, b’, c‘, be three points of K not lying on a straight line. Then the lattice A ; , with a basis (6’-a‘, c‘-a‘}, is a covering lattice of K. r f , in addition, one of the points a‘, b‘, c‘, is an inner point of K, then Ah is a covering lattice of int K. Proposition (a) is proved by taking an arbitrary C-critical lattice A and by subjecting it to a suitable transformation of the type xi = x1,x; = x2, x i = E X , +px2 f x , . Proposition (b) follows from the fact that Ah is a covering lattice of the hexagon P which is the convex hull of the points fa’, icb’, kc’. Theorem 1 can be deduced from these propositions in the following way. It is no loss of generality to suppose that K is strictly convex. Now take the lattice A , provided by (a). Consider an arbitrary point X E A , with I X , ~ < 1 and denote by a’, b’, c’, x’ the projections of a, b, c, X, respectively, onto the plane x j = 0. The three points a’, b‘, c‘ do not lie on a straight line, as otherwise A , would not be admissible for C, and c’ is an inner point of K. Therefore, by (b), there are integers u1, u2 such that
x ’ - u l ( b ’ - a’) -u2(c’- a’) E int K . Now the point x-ul(b-a)-u2(c-a)
x-u,(b-a)-u,(c-a)
has last coordinate x 3 . Thus Eint C.
Therefore, since A, is admissible, this point coincides with the origin. It follows that the points of A , are distributed over the planes x3 = k, k integral, and that A , is generated by a lattice Ah in x3 = 0 and some 2 d ( K ) , so that point in x3 = 1. We may conclude that d(A,) = d(C) 2 d ( K ) . Since we also have d(C) 2 d ( K ) , this proves the theorem.
a(&)
Few [32a] adapted the foregoing proof so as to obtain the following result.
Theorem 2. I f K is a bounded symmetric (but not necessarily o-symmetric) convex body in R2 containing a neighbourhood of o and if C is defined by (I), then d(C) = d ( K ) .
350
SOME METHODS
CH
5
The relation A(C) = A(K) also holds if K is the nonconvex star body Ix1x215 1, as was shown by Varnavides [32a]. But this relation is not
generally true. Thus Rogers [32a] found an example, where A(C) < 6 ( K ) . Later on, Davenport and Rogers [32a] constructed a class S of star bodies K such that the quotient d ( C ) / A ( K )takes arbitrarily small values on 2;a domain K in this class is obtained as the complement in the plane of finitely many small angular regions, with vertices at the points (k, 9k,, m),where 9 is a fixed irrational number. Woods [32a] proved that the three-dimensional sphere (xl)' (x,)' +(x3)' 5 1 has the same critical determinant as the corresponding cylinder C in R4.Further, Woods [32b] showed that a convex body K in RZwhich is symmetric with respect to the point o and the line x1 = 0 may not possess a critical lattice A' contained in a three-dimensional K*-critical lattice, where K* is the convex body in R 3 obtained by rotating K about the x,-axis.
+
+
+
32.2. Next we consider spheres. We denote the unit sphere in R" by K,, and deal with a method of Ollerenshaw [32a] to evaluate d ( K 3 ) . Let A be a K3-critical lattice. By a suitable rotation, it is transformed into a K3-critical lattice A , generated by three points a', a', a3 on the boundary of K 3 , where a' is the point (1, 0, 0), a' has the form (cos 9, sin 9, 0), with 0 < 9 1,
a3 has a positive last coordinate c, say. Since A , is admissible, we necessarily have 3. 5 9 5 3. Now let H,, denote the plane x3 = t ~ , and G the circular disc in RZ having on its boundary the points 0 , a', a'. The disc K3 n Ha has radius p = (1 - o ' ) ~ , and G has radius 0' = (2cos @)-I. Suppose that p' < p . Then the planar lattice L, with a basis (a', a'}, has three points 0,d,a' inside some circular disc (not centered at 0) of radius p . Hence, on account of proposition (b) discussed in the foregoing subsection, L is a covering lattice of the interior of this disc. This contradicts the fact that the set A , n H,, = = u 3 + L has no point inside the disc K3 n H a . Therefore, p' 2 p . Hence, CT 2 (1 -(p')'}' = (1 -4(CoS @)-']',
so that
d(A,) = d sin 8 2
( ~ ( C O S$9)'
-I>* sin $8.
0 32
35 1
SOME SPECIAL DOMAINS
The minimum of the last expression on the interval 3n S 9 3. is attained at the point 9 = +n and is equal to 2-+. So we have d(A,) 2 2-*. It is easily verified that, in fact, there exists a K,-admissible lattice of determinant 2-*. Hence, finally, d(K3) = 2-+.
(2)
Ollerenshaw [32b] proves, by a similar method, that (3)
4Kd=
+a
Chalk [32a] and Ap Simon [32a] investigate the domains K3(a) = (x:
(xi)2+(x2)2+(x3)2
5
K;(a) = {x: ( x 1 - a ) 2 + ( x z ) z + ( x 3 ) 2
1 7
1x31
s I},
a>,
respectively, and find that (0 < a < 1)
(4)
d ( K 3 ( a ) ) = 3a(3-a)*
(5)
d(K;(a)) = +(l-a2)(aJ3+(2-a2))~
(0 < a < 1).
The proof of (5) is based on a partial generalization of the theory of sec. 31.3 to bodies of the form K u ( - K ) , where K is a strictly convex body in R 3 containing a neighbourhood of 0 .
32.3. Finally, we consider n-dimensional cubes. The cube W: lxil S 1 ( i = 1, . . ., n) has critical determinant 1; this follows from Minkowski’s theorem and the fact that Y is admissible for W. It is also easy to see that all lattices A having the following property (P) are W-critical:
(P) there exists apermutation of coordinates transforming A into a lattice A’ with a basis
where the coordinates aki (k < i ) are arbitrary real numbers. But it is much more difficult to prove that there are no other Wcritical lattices In fact, this is equivalent with the theorem of Minkowski-
352
SOME METHODS
CH.
5
Haj6s stated in sec. 12.4 (each system of linear forms of the type (12.3) determines a lattice basis of the form ( 6 ) and conversely). Ollerenshaw [32c] proved that the hollow cube
w(u) = {x: ct
max lxil
i = l , ..., n
5 11
(0 < ct < 1)
has critical determinant m-", where m = [(l - cc)-']. With the help of the theorem of Minkowski-Haj6s she showed that m-'Y is the only critical lattice. Ap Simon [32b] investigated the excentric cube W ( a ) given by
(0 < ai < 1 ; i = 1 , . . ., n).
[xi-ail I1
(7)
This cube is connected with an interesting space filling body; therefore, we give some details. By the substitution xi = a i y i (i = 1, . . ., n) the cube W ( a )is transformed into the parallelotope G : lyi-lJ
i = 1, . . ., n).
(bi = a,:';
5 bi
Put H = G u (- G ) and define K as the set of points y with (8)
lyil
2 l+bi,
(i # j ; i, j = 1 , . . ., n).
5 bi+bj-2
lyi-yjl
Then K is an o-symmetric convex body. It is contained in H , and the lattice A , with a basis (b,+l, 1, * ., I), ( l , b 2 + l , . *
. .)l), . . ., (1, 1,. . .,b,+l)
is admissible for G , H a n d K. Furthermore, 2-"V(K) = (1+2C(bi-1)-1).n(bi-1) i
i
= d(A0).
These facts imply that K i s space filling and that d ( K ) = d ( H ) = d(G). The lattice A , is the only critical lattice of K, H a n d G. 33. A method of Blichfeldt. Density functions 33.1. In the remaining part of this chapter we deal with some methods
which yield lower bounds for the critical determinants of various types of rz-dimensional bodies. We begin by explaining a method which was used first by Blichfeldt [33a], in the case of n-dimensional spheres. Roughly speaking, the idea is as follows (Rankin [38a]).
9: 33
A METHOD OF BLICHFELDT. DENSITY FUNCTIONS
353
Let K,, denote the unit sphere in R" and let 6; = 6*(Kn)be the density of densest packing of translates K,+x (sec. 20.3). Consider an arbitrary packing of translates of K,. Replace the spheres in this packing by larger spheres (possibly overlapping) of fixed radius po > 1 and fill each of them with a certain amount of mass, of variable density, such that the total mass at an arbitrary point of space does not exceed 1. Then the product of the average number of spheres per volume unit and the mass of a single sphere will not exceed 1. This leads to an upper bound for :S and so to a lower bound for A(&). A precise formulation of this idea is given by
Theorem 1. Let po be a positive number and let 6 ( p ) be a density function which is non-negative and continuoui f o r 0 _r p 6 po and which vanishe$ j o r p > po. Suppose that, for each packing ( K n + x ' } ~ = = l , m
(1)
CS(l~-x'l)
1
r= 1
for all X E R " .
Then S,* 5 I-', where
Remark. The sum appearing in (1) is finite, because S(lx-xrl) = 0 if Ix-x' 1 > p o . The density of an enlarged sphere poK,+Y at a point x only depends on the distance Ix - x'l . Proof of theorem 1. Let W denote the cube lxil <= 1 (i = 1,. . ., n). Consider an arbitrary packing {K,+x') (r = 1,2, . . .) and, for z > 0, denote by n, the number of spheres K, +x' which are entirely contained in the cube z W , we may suppose that the centres of these spheres are just given by x', . . ., xnz. The concentric spheres poKn+X' ( r = 1,. . ., n,) are contained in (z+po- 1)W c ( r f p 0 ) W . Hence, (3)
~(lx-x'l) =
o
for r = I , .
. ., n,,
if x $ ( z + p o ) W .
Using (1) and (3) we find r
m
P
354
cn. 5
SOME METHODS
The last integral is equal to &6(p)d(K,,p") = K,,I, where So we have 2"(2+p0)" 2 n,ic,,I.
K,, =
V(K,,).
Hence, the quotient of the total volume of the n, original spheres K,, +xi and the volume of the cube z W containing them satisfies
n,K,,/V(zW) = n,~,,(2z)-" 6 (1 +po/z)"l-'. It follows now from the considerations of sec. 20, in particular theorem 20.5 and the definition of 6: = a*(&), that 6: 5 lim (1+p,,/z)T1 r+ m
= I-'.
This proves the theorem.
In sec. 38 we shall see that theorem 1, with a suitable choice of the function 6, leads to much better estimates for A(&) than the theorem of Minkowski. Here we enunciate a more general theorem (which will be applied in sec. 39). Theorem 2. Let K be a bounded o-symmetric convex body, with distance functionf. Let po ,(r be positive numbers and let 6 ( p ) be a density function which vanishes for p > p o . Suppose that, for each packing {K+x'}:==, ,
(4) Then one has
r
6(f"(x-xr)) 6 1
6*(K) 5 I-',
for all x E R". *
where I = j ) ( p ) d p " / " .
We omit the proof, because it is very similar to that of theorem 1 and because we deduce below a still more general result. 33.2. On account of (20.5) and (20.12), the inequality (5) implies (6)
d ( K ) 2 2-"V(K)Z.
We generalize this relation as follows (compare Rogers [41a]).
Theorem 3. Let S, T be two star bodies, with distance functions f, g, respectively. Suppose that T is bounded. Furthermore, let p, y, p o , IS be
*
For simplicity, we writefc(z) instead of {f(z)}".
9: 33
A METHOD O F BLICHFELDT. DENSITY FUNCTIONS
355
positive numbers and let 6 ( p ) ( p 2 0) be a density function vanishing for p > p o . Finally, suppose that the function 6 ( p ) does not exceed y and, more generally, that f o r any rn points x l , . . ., x m with
( 7)
f(xk-x.') 2 p
(k # 1 ; k, 1 = 1 , . . ., m),
one has m
(8)
6(g'(x-xr)) 5 y,
for a11 x
E R".
1
Then A ( S ) satisjies
Proof. If d(S) = coy there is nothing to prove. Suppose that d(S) is finite, and consider an arbitrary lattice A which is admissible for PS. Let P be a fundamental cell of A , let x be an arbitrary point of R" and let x l , . . ., x m be the points of A with g"(x-x') < po (r = 1 , . . ., m). Then (7) is satisfied and, moreover, Stg'(x-y)) = 0 for each point y E A which is not one of the points xl,. . ., xm. So, in virtue of our hypotheses,
2 6(g"(x-y))
YEA
c 6(gu(x-xr)) 6 Y*
m
=
r= 1
Of course, the number m depends on x. Integrating over P we get
The integral in the second member is equal to
Jomd(p)d(V(p'lUT))= V(T)JmS(p)dpni" 0 = IV(T). SinceA was an arbitrary PS-admissiblelattice, the assertion of the theorem follows.
If Sis bounded and convex, then the condition (7) means that the bodies $/?S+xk (k = 1, . . ., m ) do not overlap. Therefore, theorem 3 reduces to theorem 2 (with ( 5 ) replaced by ( 6 ) ) if we suppose S to be convex and take T = S, p = 2, y = 1. A particular case, of special importance
CH. 5
SOME METHODS
356
for the applications, arises if we take (10)
6 ( p ) = max (0, z'-p),
y = yo?',
z a positive constant. In this case, the last condition of theorem 3 (with
g = f) can be formulated as follows: if
points in R" such that
f ( x - x r ) < z (r = I,.
. ., m),
x, x l , . . ., xm are any m+ 1
m
Cf"(x-x') 1
< (m-yo)zu,
thenf(xk-xt) < /Ifor at least one pair of indices k, I with k # 1. One can describe a general situation in which this condition is fulfilled. This is done in the following theorem which is a generalization of theorems of Blichfeldt [38a], v. d. Corput and Schaake [40a] and Hlawka [13a].
Theorem 4. Let S be a bounded star body, with distance function f, and let y o , CT,z be positive numbers with z" > ((n/o)+I)yo{ V ( S ) } - ' . Then, f o r each lattice A with determinant d ( A ) = 1 , there exist a point x E R" and in > yo points XI, . . ., x" E A , such that (11)
f(x-x') < z (r
m
=
1,. . ., m), Cf"(x-x')
< (m-y,)~".
1
Proof. We shall apply the theorem of Blichfeldt to the function q ( x ) = ~(f"(x)) = max (0,z"-f"(x)).
(12)
As in the proof of theorem 3, we have
By hypothesis, the last expression is > yozu. Hence, by Blichfeldt's theorem and the relation d ( A ) = 1, there exists a point x E R" such that q ( x - y ) > yozu. Thus there exists a set of points X I , . . ., x mE A such that
zyPA
q(x-x')
z0
m
(r = 1,
. . ., m), C q(x-x') > yo?'. 1
The terms in the last sum are all z'; therefore, the number of these terms is greater than y o . Using (12) we get the relations (11). Applications will be given in sec. 40.
8 34
357
A METHOD O F BLICHFELDT AND MORDELL
34. A method of Blichfeldt and Mordell 34.1. In his original memoir [5a], Blichfeldt applied the theorem which now bears his name to the problem of the simultaneous approximation to n- 1 given real numbers by fractionspi/pn (i = 1, . . ., n- 1) with equal denominator pn ; in this way, he could ameliorate the result obtained by Minkowski (sec. 45.2). In fact, he applied theorem 16.2 to some special lattice of determinant 1 and a certain star body T with the property that the difference set 9 T is contained in the star body S given by (1)
1x11
. {max(lx,l,. . ., Ix,O]"-
5 7,
where y is a suitably chosen positive constant. Later on, Koksma and Meulenbeld [34a, 34b] and Mullender [34a] applied a similar method to more general approximation problems (sec. 45.2). Mordell [34a] and Mullender [34b] analyzed the underlying principles and deduced some general theorems. These theorems will be dealt with below. We consider pairs of star bodies S, T satisfying B T c S. As a consequence of Blichfeldt's theorem, we have
Theorem 1. Let S, T be star bodies iir R" with 9 T c S. Then A ( S ) 2 V ( T ) . Proof. d(S) 2 A ( 9 T ) 2 V ( T ) ,on account of (17.11) and theorem 17.6. Theorem 1 is the basis for our further considerations. The main problem is to choose T, for given S, in such a way that 9 T c S and V ( T ) is reasonably large. In the following, the star bodies S, T will always be symmetric with respect to the n coordinate planes x i = 0 (1 5 i 5 n); we put (2)
S, =SnO+,
T+ = T n O , ,
where 0, is the hyperoctant xi 2 0 (i = 1, . . ., n). In general, T will be a set in Rn given by a pair of inequalities of the type (3)
fI(x1, *
*
.) x p )
s
70
fZ(xp+l, * *
-3
x n ) 6 $(fi(xi
9
-
* -9
xp)),
where p is a given index with 1 S p 6 n - I , zo is a positive number, is a non-negative and steadily decreasing function of z on the interval 0 5 z 6 zo and fi, fz are the distance functions of a bounded o-symmetric convex body Kl in RP and a bounded o-symmetric con-
$(z)
SOME METHODS
358
CH.5
vex bsdy K2 in R n - p ,respectively. The star b3dy S will be defined by a similar pair of inequalities, with the same indexp. We deduce a formula by which V ( T )is expressed in terms of the functions I),f l , f 2 . Let V , , V2 denote the volumes of K , , K 2 , respectively. Then, for 0 < z 5 r o , the bodyf, S z has volume ol(z) = zpV, and the bDdy f2 S $(z) has volume w2(z) = $ ( T ) ’ - ~ VApproximating ~. T by a .finite number of ‘cylindric’ shells
I fl(Xl? * * < z ( k + I), f 2 ( x p +1 * x,) s (0 = (0) < ~ ( 1 )< * * * < z ( k ) < * - * < z ( f ) = z0),
$4
*)
7
’7
4w))
of volumes ( o , ( ~ ( k +l ) ) - - ~ , ( ~ ( k ) ).}w2(2(k)),we get the result that
34.2. In this subsection we deal with the case n = 2, p = 1. Let S c R2 be given by
s
5 cP(lXll), where z1 is a positive number and q ( ~ (0 ) 5 z 5 7,) is a convex, nonnegative andsteadily decreasing function of z. Let 5 be a fixed number with (5)
0<
(6)
1x11
5 < T,
717
1x21
and let
x2 = p-vx1
( p , v > 0)
be the equation of a tac-line to the curve x2 = q ( x l ) at the point (t, q(t)). We discuss some further conditions to be imposed upon rp and 5. The equation ( 6 ) may be written as
cc1x1+cc2x, = 1,
with a, = p ( - l v , cc2 = p - ’ .
In particular, cc,~+a,cp(<) = 1. Since the r8les of x l , x2 may be interchanged, it follows that it is no loss of generality to suppose that a2rp(5) 2 .), i.e. (7)
4 5 ) 2 +P. Next, let us consider the function q1 given by
9: 34
A METHOD OF BLICHFELDT AND MORDELL
3.59
This function is convex and steadily decreasing, whereas q(0) = p, = 0. Then we have
q l ( z , ) = ~(7,). Suppose that cp(zl)
(9)
cpl(-b,) 6
4F (7), this inequality implies that cp1(3tl)6 q(5); since q(5) = q l ( 5 )
By and pl is steadily decreasing, it follows that $5, 2 5, so that ql(3z1) = = p(&,). I n the following we do not assume that cp(zl) = 0, but only require that
dhl)6 3P.
(10)
We now come to the question of choosing T. Our first thought is to fit into S an o-symmetric convex body K and to take T = 3K. Such a body K can be obtained as follows. Since x2 = p-vx, is the equation of a tac-line and the function cp is convex, we have p-VT
5 q(t)
for 0
5z
z,.
By (lo), this implies that p-+vt, 5 3 p or also p-vz, 6 0. Therefore, the part of that tac-line intercepted by the axes is entirely contained in S. So the rhomb K : vlxll+ Ix2JS p is contained in S. Thus
9 T c S and [herefore A ( S ) 2 V ( T ) , i f T is the rhomb vIxl(+Ix,l 5 $p. It must be remarked, however, that we obtain the same result if we apply directly Minkowski’s theorem to the rhomb K. A less trivial result is obtained if one modifies in an appropriate way the star body T. This was done by Mordell[34a] who proved the following
Theorem 2. Let zI > 0 and let cp(z) be convex, non-negative and steadily decreasingfor 0 5 z zl.Furthermore, let x2 = p-vx, be the equation of a tac-line to the curve x2= q ( x l ) at some point (5, q(5)) with 0 < 5 < z1, and suppose that the inequalities (7) and (10) hold. Finally, let TO 2 5 and $(z) be defined by
Then,
if S
1x11 6 T O ,
V(T)-
is given by lxll 5 zl, lxzl 5 q(lxll) and T is given by 1x21 5 $(lxll), the critical determinant A ( S ) is not less than
360
SOME METHODS
CH.
5
Proof. Let x = ( x l , x z ) , y = (yl, y z ) be two points of T,. Then x,+yl 5 22, S z, since cp(ro) = t p 2 cp(+tl), on account of (11) and (10). We prove that we also have xz y z 5 q ( x 1 y,). It suffices to treat the following three cases.
+
+
5, 0 5 y , 6 5 . Then
Case 1. 0 5 x1 5
5 * ( X l ) + * ( Y l ) = CL--~xl+Yl) 5 c p ( X I + Y l ) . Case 2.0 5 x1 5 ( 5 y 1 5 zo. Then, since cp is convex and the tac-line at the point (t,cp(t))has slope -v, the increment cp(yl+xl)-cp(y,) X,+Y,
exceeds -vxl. Hence, by ( 1 l),
5
X,+Y,
*(Xl)+*(Yl)
Case 3. 5 5 x1 5 zo,
= -vxl+cp(Yl)
5
cp(Xl+Yl).
t 5 y1 5 zo. In this case,
xz + Y Z - cp(x1 + Y J 5 -cL+cp(x1)+cp(Y1)-cp(x1 + Y J
5 -cL+cp(5)+cp(Yl)-cp(s+Yl)
s
-p+cp(t)+v5
5
5
= 0.
We may conclude that x +y E S, , if x, y E T , . Now let x, y be two arbitrary points of T and let a = (a,, a z ) be the point (Ixli I y l l , lxzl + lyzI). By the result just obtained, a E S , . Moreover, z E S if lzil 5 a, (i = l , 2). Hence, x + y E S. This proves that 9 T c s.
+
34.3. Mullender [34b] derives the following n-dimensional analogue. Theorem 3. Let T ~ t,, p, v and the function cp satisfy the same conditions as in theorem 2 and let zo and II/ be defined by (1 1). Next, let p be an index with 1 5 p 5 n- 1, let f l y f z be the distance functions of bounded osymmetric convex bodies (of volumes V , , V,) irl RP, Rn-p, respectively, and let S be the star body in R" given by (12)
fl(X1
5
*
-Y
XP)
I z15
fZ(X,+l,
-
*>
x,) 5
P(fl(X1
9
* '
.Y
x,)).
Then one has
Proof. Let T be defined by ( 3 ) and let xyy be two arbitrary points of T.
9: 34
361
A METHOD OF BLICHFELDT AND MORDELL
Write x = (x’, x”), where x’ = ( x i , . . ., xP), x” = (xp+l,. . ., X n ) , and similarly y = (y’, y”). Proceeding as in the proof of theorem 2 we find that
fh’) +fl(Y’) s
Z1
Y
s cp(.W)+fZ(Y’)).
fi(x”)+fz(Y”)
Since the functionsf, ,fzare convex and cp is decreasing, it follows that
s
fl(X’+Y’)
Z19
fZ(~”+Y”)
Icp(fdX’+Y’))Y
that is x + y E S. So we have 9 T c S. The assertion now follows from theorem 1 and the relation (4). We give two applications of theorem 3. The first one is essentially due to Mordell [34a]; the original result of Blichfeldt [5a] and, according to a remark of Mullender [34a], also a result of Koksma and Meulenbeld [34a] are simple consequences of it (see sec. 45.2). Theorem 4. For 5 > 0, let the star body S,(t) be given by (14)
1x11 (max (IxzI,
lxil
- IxnI)>”-l s 1, s 2{-l’(“-l) (i = 2 , . . ., n).
Then the critical determinant A (Sl(())satisfies 1 n-1 where 6 = 1- (15) A(Sl(t)) > (1+ (1+6), n-1
(
-)
Proof. We apply theorem 3, with
P = 1, fi(x)
= yzl
IXlI,
fi(x2
.
, . ., xn)
= max (Ixzl,
= 2”t, cp(yz) = T-l’(n-1).
We put w = cp(g). The tangent to the curve c = cp(z) at the point
c = p-vyz, Clearly, w 2 *p
(t, w ) is given by
n 1 where p = __ an, v = __ an. n-1 n-1
2 t.0 and so
v, = 2 , v, = 2 n - l ,
. . ., IxnI),
cp(<)
(3
1 +p 2 cp(tzl). Next, we have
n-1
yzo =
2(n-1)
=
(-)
2(n-1)
n-l
=
(--)
n-l
5.
362
CH. 5
SOME METHODS
Hence, by (13) and the definition of $(z) (see (11)),
The first term on the right is equal to
_1 . {p -(p-2v5)"} vn
n-2
n-1 n
= -*
n
the second term is handled by substituting z = (2(1-p)/pY-l found to be equal to pn-l(l-p)-'dp
-
e(-1 n
= (n- 2 ) / ( 2 n - 2 )
+(n - 1)2ns0
n-2 n-1
In the last integral we expand (l-p)-' po = ( n - 2 ) / ( 2 n - 2 ) ,
= - n- P- 1o
p"(1 - p ) - l d p .
into a power series. Writing
p1 = p o ( n + l ) / ( n + 2 ) we find that
p"(l--p)-'dp = (n-1)2"-
and is
1 (n+l
=
1
1 __
pP;+2+
n+2
n+3
p"03+
I
...
=
n+1.
n
n+l
n+l nf2
n+l = 2np;++l
.
n-1 (n+l)(l-d
= (2po)n+'
The last expression is greater than
Collecting the results we find that
+2)(n-1)~ - (n+( r 1l)(n2 + 3n - 2 )
'
9: 34
363
A METHOD OF BLICHFELDT AND MORDELL
This proves (15).
Remark. Minkowski obtained essentially the crude result
One gets this result if one fits into Sl(5) the convex body given by max ( I x , ~ , . . ., Ix,,~) p-v(xll, of volume
Macbeath [34a] applied theorem 3, with p = 1, to a star body S = { x :f ( [ x l l , .. ., Ix,,~) 5 l}, where the distance function f has the following property: f ( x l , . . ., x,,) is steadily increasing in each of the n variables, in the hyperoctant xi 2 0 (i = 1, . . ., n), and the domain of points x with f ( x l , . . ., x n ) 2 1, xi 2 0 ( i = 1 , . . ., n ) is convex. He could easily prove
Theorem 5. Let S be a star body of the type described. Let (t,, . . ., t,,) be apoint on the boundary, with t i > 0 (i = 1, . . ., n), and let Cpiri = 1 be the equation of a tac-plane at this point; suppose that PI t1 5 n-l. Then (16)
d(S) 2
2"(P1p,
- - -/I,,(n-l)!)-'
x
where for each t ,c p ( t ) is the largest number such that the (n- 1)-dimensional simplex given by x1 = t , /3, 5, + * * * +/3,,5,,5 cp(.r), xi 1 0 (i = 2 , . . ., n) is contained in S, and where cp(~,) = 4.
-+
Mordell dealt with the star body TI : Ix1Iu+* * lxnl" 5 1, 0 < CT < 1, and the star body T,: Clxitxil * xip(S 1, where the sum is extended over all sets of indices i,, . . .,$,with 1 S p S n , 1s i,< i,< ipsn.
-
*
*
a
<
364
CH. 5
SOME METHODS
34.4. Next, we give an application of theorem 3, with arbitrary p . Let fl, f , be convex distance functions in p and q = n - p variables, respectively, and let S, be the star body in R" determined by
(17)
( f i ( ~ i 7 *
-
-7
xp)}"
*
{fi(xp+
1
7 * * *7
Xn)}"
s1
or shortly f, 6 q ( f i ) , where q ( z ) = T - ~ / (T ~ > 0). Furthermore, let 5 be an arbitrary positive number. The tangent to the curve CT = q ( ~ ) at the point (t,cp(t))is given by CT
= p-v7,
where p = ( n / q ) t - p ' q ,
v = (p/q)5-"/q.
We wish to estimate A(S,) from below. A crude result, analogous to Minkowski's estimate for d(S,(@) (sec. 34.3), is obtained as follows. The star body S, contains the convex body K given by fz+vfl 6 p. By theorem 1, this body has volume V ( K ) = pv, vz/;v(p-vT)qTp-%
because p"v-" = n"(pPqq)-'. Observing that p = ( n / q ) ( - P / q p, / v = (n/p)r we may conclude (Groemer [34a]): for each 5 > 0 the star body S, and even the star body S ; ( ( ) given by
f19fi4 5 1,
fi
(n/P)57
fi
5 (n/q)c-P'q
has critical determinant 5 2-"n"p!q!(n!pPq4)-l Vl V , . We now improve upon this result by applying theorem 3. We suppose that p S q; this is no loss of generality. We take T~ = 2"lp5. Then we have
q(t) = and so q(5) 2 * p 2 cp(+zl).
P = (n/q)t-P'q, q(+zl) = 4t-p/4
Also, c p ( 4 = 3117
if we take z0 = ( 2 q / r 1 ) ~ /Next, ~ 5 . if $(z) is defined as in (Il), with the present choices of p, v, z0 and cp(z), then
A METHOD OF BLICHFELDT A N D MORDELL
0 34
365
Applying theorem 3 we get
Theorem 6. Let S2 be given by (17), with given convex distancefunctions fi,fz. Suppose that p 5 q = n - p . For 5 > 0, defiote by S 2 ( ( )the domain in R" given by
Then A(S2(5)) 2 V , V 2 u p , q ,where V i is the volume of the body fiS 1 (i = 1,2) and where up,qis the last member of (18). It can be shown that up,q> 2-"n"p!q!(n!pPqq)-'and that a,,,¶ Specializing theorem 6 to the case that fl(X1
,
* *
f2(xp+13
*,
x p ) = 1x11+ -
* * .,xn)
=
*
=
- + bPL
Ixp+lI+
* * .
+Ixnl,
one gets a result of Koksma and Meulenbeld [34b]. Some more special cases are dealt with by Mullender [34c]. 34.5. The foregoing theorems lead to interesting results in the theory of diophantine approximation, by the following reasoning. The number 5 appearing in the enunciations of theorems 4 and 6 is an arbitrary positive number. The lower bounds given for A(Sl(5)) and A(S2(5)) do not depend on this number 5. Letting 5 run through a suitable increasing sequence of positive numbers 5, we easily obtain the following
Corollary. Let /Ibe a positive number. Then each lattice with determinant
(
d(A) 5 1 + nlJ1(l+6),
+)
n+2
6 = (1-
,
366
SOME METHODS
CH.5
has infinitely many points in the domain
(20)
lxll - {max ( I X , ~ ,. . ., 1xnl))”-‘ 5 I, lxil 5 p
(i = 2 , .
. ., n).
Likewise, if p, q, fi,f 2 , V 1, V , and have the same meaning and satisfy the same conditions as before, then each lattice A with determinant d ( A ) < V 1V2Q ~ has , ~infinitely many points in the domain
(21)
(fl(x19
* -3
xp)}”
*
{.f2(xp+1,
. - * s Xn)}‘ 5
1,
f 2 ( ~ p + l ,
*, X n )
5 B.
The first assertion of the corollary also holds with the condition = 2, . . ., n) replaced by lxll 5 p. We observe that the inequalities (20) are a special case of the inequalities (21). Thus, apart from the numerical constants involved, the first assertion of the corollary is contained in the second assertion. The argument by which we proved the corollary can be looked upon from the point of view of the theory of automorphic star bodies (secs. 26 and 28). Actually, the star body S , defined by the inequality (17), with a given index p and given convex distance functions fi,f,, is fully automorphic in the sense of definition 26.3, with respect to the group of automorphisrns x’ = ax given by
lxil 5 fl (i
XI= o x i (i
=
1 , . . . , p ) , xi = o-PI4 xl (i = p + 1 , .
. ., n)
(w > 0).
It is of the finite type because, for some positive constant y, it is contained in the star body lxlxz * * xnJ5 y which is of the finite type. Moreover, S, is generated by the star body given by (21), for arbitrary p > 0. Therefore, the second assertion of the corollary is a consequence of the estimate A(&) 2 Vl V Z c c p , , and the first clause of theorem 28.7. In a similar way, the first assertion of the corollary follows. For further details we refer to sec. 45.2.
-
34.6. We recall that theorem 3 is a generalization of theorem 2 to the case of n-dimensional bodies. A generaIization of a different type was given by Mordell [34a] who considered star bodies S:
(22)
Ixnl
5 d l x l I J*
*
*r Ixn-1I)y
where q ( x l , . . ., xn- 1) is non-negative, convex and steadily decreasing in each of the variables, in the hyperoctant x i 5 0 (i = 1, . . ., n). For such star bodies he proved
§ 35
361
A THEOREM OF MACBEATH
Theorem 7. Let (tl, . . ., C;,) be a point on the boundary R of S, with t i > 0 (i = 1, . . ., n). Let Cy='=l xi/ui = 1 be the equation of a tac-plane to R at the point (tl,. . ., t;,,). Suppose that <,,/a,, 2 3. Then one has (23)
d(S)
2 alaZ
-
*
cr,/n!+2"1/,
where V is the volume of the domain
xi 2 ti ( i = 1 , . xn
5 r~(x1, *
n
. ., n),
C xi/cli 2 3, i= 1
xn-1)-3an-
The proof is very similar to that of theorem 2. Mullender [34b] constructed a star body T, with 9 T c S, which for a special class of domains S leads to sharper results. If S is the domain lxlxz * x,,l S 1, then this star body T is the union of all bodies T(i, , . . ., ip):
- -
{p-'(IxilI+
*
.
+IXi,I)+~P-ln)PIXi,+l
*
xi,!
5 1,
(xik!
21
( k = p + 1 , . . ., n),
wherep is any index with 1 5 p 5 n- 1 and (il,. . ., in)is a permutation of the set (1, . . ., n). Also, Mullender [34a] improved slightly upon theorem 2 and 3 by making a more favourable choice for the function +. 35. A theorem of Macbeath 35.1. In chapter 7 we shall deal with Minkowski's theorem on the product
of two iahomogeneous linear forms. A proof of this theorem by Macbeath [47a] consists in the construction of a sequence of points x' = (xlr, xzr), all belonging to a given grid r, such that (xlrxZr[ decreases rather rapidly. The construction is continued until one gets a point X with lxlrxZrl5 p, where p is a certain positive constant. Elaborating this idea Macbeath [35a] found a general theorem on grids which, in a number of cases, can be used to estimate from below the critical determinant or the inhomogeneous determinant of a given domain. This theorem reads as follows.
Theorem 1. Let r = z f A be a grid and let H be a convex body which is pos.sibly unbounded, but has n independent tac-planes. Suppose that M = H n r is not empty. Then there exists a point x E M such that the o-symmetric convex domain ( H - x) n (x - H ) has critical determinant
368
SOME METHODS
CH.5
5 d(I') = d ( A ) and therefore has volume (1)
V ( ( H - x ) n ( x - H ) ) g 2"d(A).
We reproduce the simple proof of this theorem given by Rogers [35a 1.
Proof of theorem 1. We call a point x E M an exterior point of M if there exists a closed half-space F bounded by a hyperplane P such that x E P, M n F = 1x1.
(2)
First we prove that M has exterior points. Let PI, . . ., P, be n independent tac-planes to H. It is no loss of generality to suppose that o is the common point of these tac-planes. Let G denote the convex cone containing H which is bounded by PI, . . ., P,, . Take a point z in the interior of G and a hyperplane P containing o such that P has only the point o in common with the cone G and does not contain a point # o of the lattice A. Each point of G belongs to some hyperplane Az P, A 2 0. Furthermore, each such hyperplane cuts off a bounded portion of G and does not contain two distinct points of r. Hence there exists a smallest value of A such that Az+P contains a point x of M = H n r. Clearly, this point x is an exterior point of M. Next, we prove that each exterior point of M satisfies the assertions of the theorem. Let x be such a point. Take F and P such that ( 2 ) holds. Then ( M - x ) n ( F - x ) = {o} and so o is the only point of I'-x = A in the set ( H - x ) n ( F - x ) . In other words, A is admissible for the set ( H - x ) n ( F - x ) . It is also admissible for ( x - H ) n ( x - F ) . Since ( F - x ) u ( x - F ) = R", it follows that A is admissible for ( H - x ) n n ( x - H ) . From this the assertions of the theorem follow.
+
Remark. If H is closed, then, clearly, the set ( H - x ) n ( x - H ) has critical determinant < d(A) and so (1) holds with the inequality sign. 35.2. Macbeath gave the following applications of this theorem.
I. H is the cube [ x i [ 5 I ( i = 1, . . ., n) and r = A is an arbitrary homogeneous lattice with determinant d(A) < 1. The intersection H n A is not empty, and ( H - x ) n ( x - H ) is a parallelepiped of volume 2"nY(1 - IxJ) (x E H ) . So we have the result that each lattice A with determinant d(A) < 1 contains a point x such that
5 35
369
A THEOREM OF MACBEATH
fi (1 - Ixil) < d(n)*
(3)
1
Clearly, x #
0,
because d(A) < 1.
11. H is the hyperoctant xi > 0 ( i = 1 , . . ., n ) and r is arbitrary. Now, for X E H, ( H - x ) n ( x - H ) is a parallelepiped of volume 2"x1x2 x,; theorem 1, together with the remark, leads to the following result of Chalk [50a].
-
Theorem 2. Each lattice and, more generally, each grid r contains a point x with x i > 0 ( i = 1, . . ., n), x1x2 * x,, 5 d ( r ) and also a point y with y i 2 0 ( i = 1 , . .)n), y 1y 2 * * * y,, < d ( r ) .
- -
.
111. H i s the convex domain (4)
xi > 0 (i = 1, .
. ., n),
xlxz
x,
* * *
2a
(a
> 0 constant).
We consider the set (H-x) n ( x - H ) , for a given point x E H. By the transformation y i = xiy; ( i = 1, . . . n; x = (xl . . .,x,,) fixed) it is transformed into the set (5)
--
where we have put 9 = a//?, /? = xlx2 x,, (so that 0 < 9 If q(9) denotes the volume of this set, then
5 1).
with p = x1x2 * * x, , 'V((H-x) n ( x - H ) ) = pq(a/j?), By studying the behaviour of q(9) for 9 -+ 1 one can prove that there exists a constant K, > 0 such that prp(a/p) 5 2" if a is sufficiently large and p = a(1 + (K,,a)-'/("+')). So one has /? 5 a(1 + ( i ~ ~ a ) - ~ / ( " + ~ ) ) if a is sufficiently large and V( (H-x) n (x-H)) 5 2". Applying theorem 1 and using homogeneity considerations one gets Theorem 3. An arbitrary grid r contains a point x with
d(r)
xlx2
--
X,
5 a(i+(~,a)-~/("+l))d(r),
provided a is positive and sufficiently large.
Macbeath also applied theorem 1 to the sets X1-(X2)Z-
(x1)2-(x2)2-
*
-
*
* -(XJ2
-
-(XJ2
2 0; 5 0.
370
CH. 5
SOME METHODS
35.3. Macbeath's original proof of theorem 1 is more complicated, but yields some extra information. Therefore, we reproduce it here. Let r, H satisfy the conditions of theorem 1. Consider the function 'p defined by (6)
cp(x) = V ( H , x ) = v ( ( H - ~ ) n (x-H))
(x E H ) .
It is non-negative on H and unbounded on each unbounded convex subset of H. By applying the theorem of Brunn-Minkowski to sets ( H - x ) n (x- H ) one finds that {'p(x)}"" is a concave function 0f.x and that this expression, when restricted to a closed bounded convex subset of H, attains its maximum at one point only. Now let xo be an arbitrary point in the interior of H. Let G(xo) denote the set of points X E H with p(x) 2 p(xo). Then G(xo) is a convex set, and xo lies on the boundary of this set. We take a tac-plane P to G(xo) at the point xo and denote by H' the half-space which is bounded by P and does not contain G(xo). Then the function cp is bounded on H n'H' and therefore H n H ' is bounded. So there is a tac-plane P' to H which is parallel to P and is contained in the interior of the half-space H'. Let it be given by P' = P - a . We now put
(7)
K = H n H', K' = 3 ( K + x 0 ) , K, = H n (H'+a).
Thus K is the part of H cut off by the hyperplane P , K, is the part cut off by P + a , and K' is obtained from K by a dilatation, with a factor +, with respect to the point xo. From the definition of K, K, and the function cp it is clear that (8)
q(H,x ) = q ( K 1 ,x)
if x E K .
We investigate, in particular, the quantities $ ( K , ) and z(K,) given by
(9)
$(Ki) =
XSK
q(Ki 3 X)Y 7(Ki) = max q(Ki Y x)* x E K\K'
We have z(K,) c $(K,) because the first maximum in (9) is attained at the point x = x o only. We now take into consideration the collection of all possible sets K , and use a compactness argument. The set Kl is bounded and convex. The hyperplanes P - a , P + a are tac-planes to Kl, and xo E P is an inner point of K,. The function cp(K,, x) takes its maximum on the set K at the point xo only. Considering parallelepipeds containing Kl we see that there exists an affine
0 35
371
A THEOREM OF MACBEATH
transformation A which sends x o into o and transforms K , into a convex set L , = A K , with the following properties 1". the hyperplanes x1 = f 1 are tac-planes to L , ; 2". the set L , is contained in the cube lxil S 1 (i = 1,. . ., n ) and has volume V ( L , ) 5 l/n!; 3". o is an inner point of L , and the function q ( L , , x ) attains its maximum on the set { x : x E L ,, x , S 0 } at the point o. The convex bodies L , possessing the properties 1 2", 3"form a compact metric space* in the metric 6 given by (25.3). The functionals $ and z defined on this space are continuous in the metric 6 and positive. Now the quotient T(K,)/$(K,) is left invariant under an affine transformation. Since this quotient is always less than 1, if follows that there exists a positive constant A,,< 1 only depending on n such that T ( K , )< l,,$(Kl). Thus we have O,
(10)
q(H,x) 5 1,,q(H, xo)
for x E K\K'.
This relation holds, with a fixed constant A,,< 1, for each point x o E H and a corresponding pair of sets K , K'. The proof of theorem 1 is now completed as follows. By the theorem of Minkowski-Hlawka (sec. 19). we have
(11) ~ ( ( H - x n ) ( x - H ) ) Iv ( ( H - n ~()x - H ) ) = q ( ~x), (x E H ) . We wish to show that A ( ( H - x ) n ( x - H ) ) 5 d ( T ) for some point x E M . Take a point x1 E M and suppose that A ( ( H - x ' ) n (x' - H ) ) > > d ( r ) .Then A has a point # o in H - x1 and so r has a point # x1 in H , because XIE r. We also have q ( H , x') > 0, by (1 1). Hence, x1 is an inner point of H . Taking xo = xl, in the definition of H ' , we have that r has a point # x1 in K = H n H'. Then there is a point of r in K\K'. Let xz be such a point. It is a point of M and, by (lo), it satisfies the relation
5
.')If it happens that A ( ( H - x 2 ) n ( x 2 - H ) ) > d(T), then we repeat the process. And so on. At each step we gain a factor 1,. Therefore, after a finite number of steps, we get a point x' E M with d((H- x') n (x' - H))I 5 d(T). This proves the theorem. V(H, x')
I n q ( ~ 7
As Rogers [35a] remarked, the considerations of Macbeath contain implicitly the following result.
*
See the considerations of sec. 25.3.
312
CH. 5
SOME METHODS
Theorem 4. There exists a positive constant 1, < 1 only depending on n which has the following property. Ifr, H satisfy the conditions of theorem 1 and i f q ( x ) is dejined by (6), then each point x E H n r with cp(x)
<1 '; inf ~ ( z ) zeHnr
is an exterior point of H n r. (Such points x exist if and only if the infimum is positive). Indeed, if x were not an exterior point of H n r and if H ' were defined as absve, then H n H' would contain at least one point x' # x of
r
with cp(x')
5
An(p(x) < inf cp(z). zeHnT
Rogers investigated also under what circumstances the set H n r has at most finitely many exterior points. Macbeath's method may be used to give an alternative proof of the theorem of Cebotarev (sec. 49.2).
36. Comparison of star bodies in spaces of unequal dimensions 36.1. Hermite was the first who derived an upper bound for the homo-
geneous minimum of an arbitrary positive definite quadratic form (sec. 5.1). As Mordell [36a] remarked, Hermite essentially proved a certain inequality connecting the absolute homogeneous minima of two positive quadratic forms in n and n- 1 variables, respectively. Later, the method was applied to other forms, in particular by Mordell [36a, 36b, 36c] and Oppenheim [36a, 42~1.In this section we prove a general theorem exhibiting the method; it was given in a less general form by Armitage [36a]. Let S be a star body with distance functionx In the following, we shall operate with the absolute minimum A(S) rather than the critical determinant A(S) and also consider the absolute minimum %(Sn P ) of the ( n - 1)-dimensional star body S n P, where P is some ( n - 1)dimensional linear subspace of R".For z # 0, we shall denote by P ( z ) the hyperplane which contains o and is orthogonal to z. Furthermore, for a given lattice A = AY, we shall consider the polar lattice A* = A*Y (see sec. 3.5). We recall that {%(s)]-"= A(S). In particular, 1(S) = 0 if S is of the
§ 36
COMPARISON OF STAR BODIES IN SPACES OF UNEQUAL DIMENSIONS
313
infinite type. We also have (1)
A(S) = sup Al(S,A){d(A)}"".
The supremum in the right hand member of (1) is attained for some lattice A , on account of theorem 26.6. Let .4 be an arbitrary lattice in R". Let b be a primitive point of the polar lattice A*. Then A has a basis {a', . . ., a"> such that (2)
a i * b = 0 (i = 1,. . ., n - l ) ,
a " . b = 1.
Thus A has n - 1 independent points in the hyperplane P(b). We put
L = A n P(b). (3) Then L is an ( ( n - 1)-dimensional) lattice in the (n- 1)-dimensional space P(b). As such, it has a determinant d ( L ) defined as the (n- 1)-dimensional volume of a cell of L. Taking a cell of A spanned by a cell of L and the point a" we see that the point a" has distance d(A)/d(L)to the hyperplane P(b). Hence, since b is orthogonal to P(b) and a" . b = 1, (4)
Ibl = d ( m @ ) .
We now prove (Lekkerkerker [36a])
Theorem 1. Let S be a star body of the finite type, with a group r of automorphisms, and let Po be a hyperplane through the point 0. Suppose that the following conditions hold 1". the set of points z # o with A(S n P ( z ) ) = 0 is not dense in R"; 2". for each hyperplane P containing 0, with 1 ( S n P ) > 0, there exists an automorphism Q = Sap E r with QP = Po; 3". if Sa E r, then O* E r. Then there is apoint bo on the boundary of S with bo (5)
Po; moreover,
{A(S)>.-z 5 {A(S n Po))"-*lbol.
Proof. Let f be the distance function associated with the star body S and let A be an arbitrary lattice. Let b be a primitive point of the polar lattice A*; write S(b) = S n P(b), L(b) = A n P(b). From the definition of the minima A,(&
(6)
A ) , Al(5'(b), L ( b ) ) it is clear that
w,4 5 Al(S(% W)).
374
SOME METHODS
CH. 5
We shall estimate Al(S(b), L(b)).Thereafter, by an appropriate choice of b, we shall obtain a good estimate for Al(S,A ) . If it happens that A(S(b)) = 0, then necessarily A1(S(b), L(b)) = 0 (see the relation (17.17)). Therefore, it is no loss of generality to suppose that A(S(b)) > 0. Then, by 2", there exists an automorphism 52~r such that BP(b) = P o . We have Po = P(b*) if we take b* = Q*b.
(7)
The point b* belongs to Q*A* = (an)*.Applying (4) to the point b* and the lattice !& and I observing that SaS(b) = S n P o , QL(b) = Q A n P , we find that
(8)
Al(S(b)J(b)) = = I , ( S n p0,QA n P,) 6 A(S n P , ) { ~ ( Q An Po)}"("-') = = A(S n P0){Jb*Jd(S2~)}'""-'' = A(S n Po){~b*~~(~>}'~~"~''.
From (7) it is clear that b* # 0. We prove that f ( b * ) > 0. Suppose that f ( b * ) = 0. Let z be a point # o with A(S n P ( z ) ) > 0. Then, by 2", there exists an automorphism 0,E r such that Q,P(z) = P o . We have b'f = d * z , with some ct # 0, because P ( z ) is orthogonal to z and Po is orthogonal to b*. By 3", C2; is an automorphism of S . Hence, f ( b * ) = lalf(z) and s o f ( z ) = 0. Applying 1" we see that the last relation holds for all z in some cone with vertex at 0. Thus S contains some infinite cone, which cmtradicts the assumption that S is of the finite type. This proves that f ( b * ) > 0. We now put bo = ab*, where CI > 0 is so chosen thatf(bO) = 1. Then
lb*I = f (b*) lbol =f(b*)lbol = f ( b ) l b O / ,
f (bO) because C2* is an automorphism of S. We substitute this in the last member of (8). At this stage, we make a definite choice for the point b. We recall that b was required to be a primitive point of A*. Accordingly, by the definition of A,(S, A*), we can choose b in such a way that f ( b ) < A,(S, A*)+&, where E is an arbitrarily chosen positive number. Using ( 6 ) and (8) we find that &(S, A ) < A(S n Po){(Al(S,n*)+~)Ib~ld(A))"("-') =
6 A(S n Po){~(S)(~(l"))""+~)"("-')(~bo~d(/l))''("-').
(j 36
COMPARISON OF STAR BODIES IN SPACES OF UNEQUAL DIMENSIONS
Since this is true for each conclude that
E
375
> 0 and since d(A*) = { d ( A ) } - ' , we may
Al(S,A ) 6 A(S n Po){A")~bo~(d(A))l-l'"}'/'"-''= =
A(S n Po){"S)[b0~}"'"~''{d(~)}''".
The lattice A being arbitrary, it follows that A(S) S A(S n PO){"S)lbo~}'l'"-''.
This proves (5) and completes the proof of the theorem.
36.2. We make a number of remarks. Remark 1. Similar considerations hold if instead of Po, there are finitely many hyperplanes P(b'), . . ., P(b'), where b', . . ., b' lie on the boundary of S, such that each hyperplane P through 0,with A(S n P ) > 0, can be transformed by a suitable automorphism QETinto one of the hyperplanes P(b'), . . ., P(b'). In this case, instead of ( 5 ) , one gets the estimate
(9)
{A(S)}"-z 5 max {{A(S n P(bk))}"-'Ibkl). k = 1,.
. ., r
Remark 2. For the applications it may be convenient to replace the requirements lo, 2" by the following two conditions (confer remark 1): 1'. for each point z # o with f ( z ) = 0 m e has A(S n P ( z ) ) = 0 . 2'. i f z is apoint withf( z ) = 1, then there is an automorphism transforming z into one of r givenpoints b', . . ., b' on the boundary of S. It is clear that, in the case r = 1, the conditions 1' and 2' imply the validity of 2". Furthermore, it follows from the proof of theorem 1 that 1" is not needed if I f , 2' and 3" hold. Remark 3. Suppose that, with the notations of remark 2, none of the hyperplanes P(b'), . . ., P(br) contains a given coordinate axis, i.e., that none of the points b', . . ., b' lies in a given coordinate plane x i = 0. For k = 1, . . ., r, denote by P k the i-th coordinate of bk and by the domain in the hyperplane xi = 0 which is the projection of S n P(bk) onto this plane. Then one has, for k = 1, . . ., r,
316
CH. 5
SOME METHODS
On account of these remarks we have the following variant of theorem 1.
Theorem 2. Let S be a star body of theJnite type, with distance function r of automorphisms. Suppose that the conditions l', 2', 3" and the hypothesis of remark 3 hold. Then, if the numbers p k and the projections are defined as above,
f and with a group
The method can also be applied if S does not admit a group of automorphisms fulfilling the conditions of theorem 1 or theorem 2, although the result is less useful. Consider the function g defined by
(11)
g(z) =
{A(s n P ( Z ) ) ) " - ' ~ Z ~
(z
+ o),
g(o) = 0.
If this function is continuous, then it is the distance function of some star body T. We prove that
{n(s)}ii-l 5 A(T).
(12)
Let A , b, L ( b ) have the same meaning as in the proof of theorem 1. Then
A(&
A)
5 A(s n P(b), W ) )5 6 A(S
n P(b)) {d(L(b))}"("-" = {g(b)d(A)}'""- '1.
The point b E A* can be chosen such that g ( b ) is not much larger than 1(T,A*)
s A(T)(d(A*)}"" = A(T){d(A)}-l?
So we have A(S, A ) 5 { A ( T ) } ' ~ ~ " ' ~ { d ( A ) } 'Since ~ " . A is arbitrary, the relation (12) follows.
36.3. We give some applications. First, we take as S the unit sphere K,. Then the condition 1" of theorem 1 is trivially true and the conditions 2", 3" hold if we take as r the group of all rotations about 0. So we have the following result (Mordell [36a], Oppenheim [36a])
(13)
A(K,)
5 {A(~"-~)}(n-1)'(n-2).
Next, we consider the star body So given by lxlxz * * * x,J 5 1 (Mordell [36b, 36~1).Let r be the group of all automorphisms xi = mixi
(i = I, . . ., n) (wi real, w1 w2
-
* *
on=
k 1).
3 36
COMPARISON OF STAR BODIES IN SPACES OF UNEQUAL DIMENSIONS
377
We prove that the condition 2' holds. Let z be a point # o with * * z,, = 0 ; without loss of generality we may suppose that z1= 0 and that z,, # 0. Then P(x) has equation z,x,+ * * * +z,,x,, = 0 and the projection of Son P ( z ) onto the plane x,, = 0 is given by
zlz2
(14)
lx1xz
--
*
x,-,(Cr,x,+
--
*
+Cr,,-'x,,-')~
s 1,
where cti = zi/z,, (i = 2, . . ., n- 1). If fi = Ict,l+ * * * +Ict,,-l( and > 0 is arbitrarily large, then the domain (14) contains the parallelepiped given by z
1x11
5 pP,
lxJ
5z
(i = 2 ,.
. ., n-1).
This parallelepiped has volume 2"-'P-'z. It follows that the domain (14) is of the infinite type. The same is true for So n P(z). Hence, condition 2' is fulfilled. . The conditions 1' and 3", with r = 1, are easily verified. We now take b' = (I, 1 , . . ., 1) and project the intersection Son P(b') onto the x,,-axis. Applying theorem 2 we get the following result.
Theorem 3. I f S , is given by lxlx2 * * x,,l 5 1 and i f S , , - , is the (n- 1)dimensional star body gioen by Ixl x2 * x,, - (xl + . * . + x,, - 5 1, then
n:='
Armitage [36a] gave a similar result for a star body given by an inequality of the type {Fi(x)}"'5 1; here the ni are positive integers with ni = II and Fi(x)is a sum of nisquares (xj)'. He deduced a more complicated inequality for the critical determinant of the domain (x')2{(x2)2+ . * . ( x $ } 5 1. Some further applications will be mentioned in secs. 41.4,42.3,42.4 and 45.4 (Mordell [41b, 42g], Oppenheim [42c], Mullender [45a], Davenport [45cI),
1
+
36.4. In the final part of this section we deal with an entirely different
method of comparing bodies in spaces of unequal dimensions. This method leads to upper bounds for the inhomogeneous minimum p(K, A ) of a given convex body K with respect to a given lattice A and has applications in the theory of diophantine approximations (secs. 45, 51). It goes back to Mordell [24a] and is often called the method of the additional variable.
378
cn. 5
SOME METHODS
The method is illustrated by the following theorem (Mahler [13a], Hlawka [13~1).
Theorem 4. Let K be a closed bouaded o-symmetric convex body in R". Put (16)
p = [(2/1,)"V-']+1,
where A, = l , ( K , Y ) and V = V(K).
Then for each point z E R" there exist a point u E Y and an integer k such that (17)
1 I;k 5 p - 1
and u + k z E ~ K , where p = 2(pV)r"".
Remark. By Minkowski's theorem, AlV = V(A,K) S 2". Hence (2/Al)"V-l 2 1, so thatp is an integer >= 2.
-
Proof of theorem 4. We havep > (2/A1)"V-', and so p < 2 (2/Al)-' = = A,. Let now pl be any positive number with p < pl < A, and let K' be the (a-t-1)-dimensional body consisting of the points x' = = (xi,. . ., x, xn+1 ) with (18) X+Xn+lZEP1K, lXn+ll 5 P . Here z is a fixed point in R" and x is the point ( x l , . . ., x,,) E R". The body K' is a skew cylinder with height 2p and with an axis passing through the origin and the point ( - z , , . . ., - z n , l), where z l , . . ., z,, are the coordinates of z. We apply Minkowski's theorem to the (n 1)-dimensional body K'. It is clear that K' is convex and o-symmetric. Furthermore,
+
V(K') = 2pV(p1K ) = 2pp; > 2pp"V = 2"+l. Hence, there exists a lattice point u' = (u,, such that, if u = ( u l , . . ., u,,),
. . ., u,, u,+
# (0,.
. ., 0,O)
I U , + ~ I 5 p-1. Since such a lattice point U' exists for all p1 > p and since Y is bounded U+Un+1ZEpiK,
lu,+lJ < p
and
SO
and closed, it follows that there also exists a lattice point u' = = (ul,. . ., u,, un+,) # o with
(19)
U+Un+lzEPK,
IUn+11
6 P-1.
Suppose that un+ = 0. Then u # o and u E pK. But this is impossible, because p < 1 , . So we have un+, # 0. For reasons of symmetry, we may suppose that un+ 0. This proves the theorem.
,=-
4 36
COMPARISON OF STAR BODIES IN SPACES OF UNEQUAL DIMENSIONS
379
Using theorem 4 we can easily deduce an upper bound for p ( K , Y ) and thus obtain a new proof of theorem 13.2 (Hlawka [ 13~1).The argument is as follows. Let K have distance functionfand let z o be an arbitrary point of R". We wish to estimate minf(zo-u). It is no loss of generality to suppose that this minimum is attained for u = 0. Put z = (2/p)z0, where p is given by (16). Take a point u and an integer k for which the relations (17) hold. Then we have f(+PZ)
5 f ( U + 3PZ) 5 f ( U + k z ) +f((+P - k ) z ) s
hence
f(4P) 5 3PP
=
P(PW/".
By the arbitrariness of z o , it follows that
P ( K Y)5
p(pV)-"".
This is slightly stronger than (13.7), because (pV)-"" <
+A,.
Schneider [13a] deduced theorem 13.5, in the special case that H I = H , and these bodies are o-symmetric and convex, by considering an (n+ 1)dimensional body of the form x + x , + ~ z E I X , H, + ~ ~~ , 5 2. A further application of the method is a proof by Mordell [51a] of (the first assertion of) theorem 51.6.
NOTE ON CHAPTER 5 As we indicated in the preface progress in the geometry of numbers was by no means uniform. In several areas modern as well as classical ones new methods were developed. But there were no significant contributions to the subject matter of Chapter 5, viz. classical methods for the determination or estimation of critical determinants. For this reason we do not insert a supplementary section on 'Methods'. For references to several new methods in this area we refer to sect. iii, vi, xi and xiv.