Chapter 7 Inhomogeneous Forms

CHAPTER 7 INHOMOGENEOUS FORMS

In this chapter we deal with inhomogeneous minima. In particular, we treat indefinite binary quadratic forms and, more generally, products of linear forms, The title of the chapter wil1 be explained in the first section. Sometimes we shall be concerned with relations between homogeneous and inhomogeneous minima. 46. Inhomogeneous minima of forms 46.1. In this section we recall some definitions and simple properties

related to the concept of an inhomogeneous minimum. We also express various quantities in terms of forms. Let S be a given star body in R", with distance functionf, and let A be a lattice in R". According to definition 21.2 and theorem 23.1, the covering constant and the inhomogeneous determinant of S are given by

(1)

f(S) = sup {d(A):A a covering lattice of S }

(2)

d'(S) = inf {d(A):A not a covering lattice of S}.

We remark that, since o is an inner point of S, there certainly exist covering lattices of S. It may happen that each lattice A is a covering lattice of S ; if this is the case, then we put d'(S) = co. The quantity f ( S ) is positive (possibly infinite); the quantity d'(S) may be equal to zero. We always have (3)

d'(S)

s qs).

The two quantities f ( S ) , d'(S) are related to the absolute inhomogeneous minima of S. These minima are given by

(4)

p0(s) = inf { p ( ~A, ) : d ( ~=) I} = infp(S, A>{~(A)}-""

(5)

pO(S) = sup {p(S, A ) : d(A) = l} = sup p(S, n){d(A)}-"",

where

550

INHOMOGENEOUS FORMS

(6)

p ( S , A ) = inf {a: r~ > 0, A a covering lattice of as).

CH.

7

We have

r(s)= {p0(S)}-n7

(7)

"(')

= {po(S)}-n

(see theorems 21.2 and 23.1). It is trivial that p o ( S ) 5 p O ( S ) ,in accordance with (3). The inhomogeneous minimum p ( S , A ) is always finite. It can be expressed in terms of the distance function f (and the lattice A ) , as follows. For z E R" put p(S, A , z ) = inf { f ( z - x ) : x E A } .

(8)

Then, according to theorem 21.5, $ ( z ) = p ( S , A , z ) is upper semicontinuous and periodic mod A , and we have p ( S , A ) = max { p ( S , A , z ) : z

(9)

E

R"}.

Occasionally we write p ( S , r )instead of p ( S , A , z ) , r denoting the grid A+z. In the following, instead of p o ( S ) , p o ( S ) , p ( S , A ) , p ( S , A , z ) , we usually write po(f), p o ( f ) , p ( f , A ) , p(f, A , z ) , respectively. Let F be a homogeneous form of degree D > 0 related tofby the formula IF(.)/ = = { f ( x ) ) " (see definition 37.1). Put (10)

=

{h(f)}"7

=

{p0(f))"'

We call these quantities the absolute inhomogeneous minima of F. By (7), we have

p o p ) = {r(s)}-"/", p o p ) = {d'(s)}-"/".

(11)

Further, we put (12)

P(F7

4=

(PU-7

A>>", P(F7 -4z )

=

{P(f,A , z)}" ( A a lattice;

z

E R").

The first quantity is called the inhomogeneous minimum of F with respect to A . We have (13)

p ( F , A ) = max { p ( F , 4,z): z

(14)

p(F, A , z ) = in€ { I F ( x - z ) l : x E A } .

E

R")

5 46

INHOMOGENEOUS MINIMA OF FORMS

55 1

For fixed z, the function G ( x ) = IF(x-z)l is called an inhomogeneous form. The points x - z , x E A , constitue the grid A - z . It may occur that, in spite of (13), there exist points z such that IF(x-z)l > p(F, A ) for each point x E A (Inkeri [47d]). We prove Theorem 1. Suppose that to each point z E R" one can find a point x with J F ( x - z ) l 5 p(F, A ) . Then there is apair ofpoints z , x satisfying (15)

IF(x-z)l

=

p(F, A , Z )

=

P(CA ) .

If the hypothesis of the theorem is fulfilled, then we say that the minimum p(F, A ) is attained. Proof of theorem 1. Put p = p(F, A ) . If z is a point with p(F, A , z ) < p, then there are points x E A with IF(x-z)l < p. Suppose that this is also true for the points z with p(F, A , z ) = p. Then, since F is continuous and a (closed) cell of A is compact, there is a positive number q such that for each point X E R" there exists a point X E A with IF(x-z)l < p-q. This contradicts the definition of p = p(F, A ) . So there is a point z with p ( F , A , z> = p for which there does not exist a point X E A with / F ( x - z ) l < p. From this the theorem folIows. In general, p(F, A , z) is not a continuous function of z . It may have isolated values in the following sense. Let P be a cell of A . Then there may exist a finite sequence of points z l , . . ., zk E P and a number q > 0 such that P ( F , A , z ) < min p ( ~A, , z i ) - q i = l , ..., k if z E P and z # zi ( i = 1, . . ., k). If this is the case, then the values p(F, A , z i ) are called isolated minima of the inhomogeneous form F ( x - z ) . In a few cases it has been possible to trace an infinite sequence of isolated minima, e-g., for the norm forms N , and N , (sec. 47.5). Likewise, for given F, the inhomogeneous minimum p ( F , A ) is, in general, not a continuous function of A. Actually this minimum, or rather the 'normalized' minimum p(F, A ) {d(A)}-e'", may have isolated values, as in the case of homogeneous minima (sec. 43). These values, too, are called isolated minima. Such an isolated minimum has been found in the case that F is an indefinite ternary quadratic form (Davenport [49h]).

552

INHOMOGENEOUS FORMS

CH.

7

Of course, in the expression p(F, A, z), we may also vary A and z at the same time. A related quantity having isolated values is the function p.'(S, a ) discussed in sec. 47.7. On the other hand, we have the following theorem which is an analogue of theorem 25.6 concerning homogeneous minima (compare Godwin Wall. Theorem 2. Let F be a homogeneous form. Let {A,} be a sequence of lattices converging to a lattice A and let {z'} be a sequence of points converging to a point z. Then p(F, A, z ) 2 lim r + m sup p(F, A,, 2'). Proof. Take bases A , A, of A , A, (r = 1,2, . . .) such that A , -+ A as r + 00. Let x = Au be an arbitrary point of A. Then, by the continuity of F, IF(x-z)l = lim IF(A,u-z')l 2 lim sup p ( F , A , , z'). r+m

r-

00

From this the assertion follows. 46.2. Again let F be a homogeneous form of degree 0 > 0. For any non-

singular linear transformation A of R", let FA be defined by FA(x)= F(Ax) ( x E R"). Then, as in the case of homogeneous minima,

(16)

p(FA 7

z ) = p ( F ~A x

Az)

(.

R")~

and so p(FA 7 y )

(17)

=

p ( F , AY)*

Applying (lo), (4) and (5) we get (18) po(F) = inf p ( F A , Y)ldet AI-"'", po(F) = sup p ( F A , Y)ldet A]-"'". A

A

Also, (19)

,u0(FA) = po(F)ldet AI'"", po(FA) = po(F)ldet A!"'".

If G is equivalent to a non-zero multiple of F, then we write G

E

I;.

47. Indefinite binary quadratic forms 47.1. Let Q(x) = ct(xl)' +2px,x, +y(x2)' be an indefinite binary quadratic form, with discriminant 6 = 6 ( Q ) = /?' - ccy > 0. The simplest

§ 47

553

INDEFINITE BINARY QUADRATIC FORMS

example of such a form is P ( x ) = x1x2. It has discriminant $. An arbitrary form Q can be written as PA,where A is a linear transformation of R2 with determinant f(46(Q))+. Consequently, by (46.18), Q

In the following the star body IP(x)l = Ix1x215 1 is denoted by S. Minkowski* proved geometrically the following Theorem 1. For each Q, p(Q, Y ) 5 (l,s(Q))+.The equality sign holds if and only i f Q z P .

According to this theorem, we have p o ( P ) = 4.An equivalent statement is that p o ( S ) = f , or also d'(S) = 4. More generally, we have po(Q) = =

(ta(Q))'.

Another remark is that the inhomogeneous minimum of the particular form P is attained, i.e., for each point z E R2 there exists a point u E Y such that IP(u-z)l = I(~1-Z1)(uz-z2)1

s p(P, Y) = a.

This is easily verified. Now, by the theorem, p(Q, Y ) < (@(Q))* if not Q w P. Hence we have (see the proof of the corollary to theorem 42.1) Corollary. For each form Q and each point z E R2, there exists a point u E Y satisfying the inequality lQ(u-z)l 5 po(Q) = (+a(Q))'.

Proof of theorem 1. Let A = A Y be a lattice of determinant 1 . On account of theorem 41.2, it contains a primitive point a = (a1, a 2 ) such that la,a21 5 3, so that la1 5 1. There exists an automorphism Q of S such that lQal = 1. We put a' = Qa and A' = QA, and consider the square K with centre at 0 , with side length 1 and with sides parallel and orthogonal to the vector a'. This square is contained in the disc (xl)' (x2)' 5 4 and therefore also in the domain 3s = {x: lxlx21 5 +). Now it is clear that A' is a covering Iattice of K . Hence, A' is a covering lattice of +S. The same is true for the lattice A. Thus p(S, A ) 5 4. In the particular case A = Y we have p(S, Y ) = 3. If A does nof have the form QY, Q an automorphism of S, then the point a may be so chosen

+

*

Minkowski [GA], I, p. 342.

554

INHOMOGENEOUS FORMS

CH. I

+

that la, a21 5 and that a does not lie on one of the coordinate axes. Then the corresponding square K is contained in p S for some p < +, and so p(S, A ) < 4.From these remarks and the relation p ( P A , Y ) = p ( P , A ) = = ( p ( S , A ) ) 2 the theorem follows, at least in the case that h(Q) = $. For reasons of homogeneity, the theorem is then also true in the general case.

It should be observed that the first assertion of theorem 1 is not generally true if Q is a binary quadratic form with discriminant 0. fn this case, Q ( x ) is a complete square, say (allx,+a,zx2)2;if a , , x , + a , 2 x 2 is a multiple of a rational form, then p ( Q ) > 0. In the literature a good many proofs of theorem 1 are known. Most of these proofs are of an arithmetical nature. We mention Perron [47a], Landau [47a], Pall [47a], Mordell [47a, 47b], Cassels [47a], Macbeath [47a]. See also Koksma [DA]. Perron's method is that of the proof of theorem 7 (see below). One of Macbeath's proofs runs along the following lines. Let A be a lattice of determinant 1 and let (a, b } be a basis of A with a E S. Take an arbitrary point z E R 2 .Then one can find an integer k such that the point x = z + k b satisfies ldet (a, x)I S +. Thereafter, one can find an integer h such that the point y = x + h a satisfies Iy1y215 4. In another proof published in the same paper Macbeath takes an arbitrary lattice point uo and then, for given Q and z, he constructs a sequence of lattice points u l , . . ., ur such that lQ(u'-z)l decreases to a value 5 @ c ~ ( Q )(confer )~ Macbeath's theorem dealt with in sec. 35). Sawyer [47a] deduces theorem 1 from the following two propositions.

+

I) Let r = A z be a grid in R2 having no point on the coordinate axes. Then there is a closed quadrangle R with vertices in r,one in each quadrant, such that R does not contain other points of r. 11) If R is such a quadrangle and, in addition, r is admissible f o r S, then d ( T ) = area R 2 4.

Proposition I) is an immediate consequence of theorem 2 to be proved below. Proposition 11) is proved as follows. Let a = (al, a 2 ) , b = = ( -pl, p,), c = ( - y l , - y 2 ) , d = (a,, - 6 , ) be the vertices of R in the first, second, third and fourth quadrant, respectively. Then CY, ,a Z ,p1, . . . are positive and the products a , u 2 , PIP2, y l y 2 , 6,6, are all 2 1, because r is admissible for S. Hence,

§ 47

555

INDEFINITE BINARY QUADRATIC FORMS

(2) area R

=

+{det (a, b)+det (b, c)+det (c, d)+det (d, a ) ) =

=

Hal P z + “ Z P l +PI72

+ P Z Y l +?la,

+Yz 6,+ a 1

a2 +

6 2 4 h

On the other hand, by the theorem on lattice triangles (theorem 3.4), the triangles abc, cda, both have area +d(T).Hence R has area d(T).This proves 11). It is easily seen that the equality signs hold in (2) if and onlyif u1 = P1, = y1 = 6, and az = PZ = y z = 6, = l / a l . Thus, under the conditions of proposition I]), d(T) = area R = 4 if and ‘only if T has the form D(Y+z), where z is the point (4,S) and D is an automorphism of S. From this remark, together with propositions I) and II), the assertions of theorem 1 follow. 47.2. Let

r = A+z

be an arbitrary grid in R2.By a cell of T we mean any parallelogram R + y , where y is a point of and R is a cell of A . Delone [48a] calls such a cell a divided cell of T if it has a vertex in the interior or on the boundary of each of the four quadrants. He proves

Theorem 2. Let f‘ be a two-dimensional grid not having points on the coordinate axes. Then r has a divided cell. Proof. Let a be a point # o of A which is nearest to o and let L be a straight line passing through points of A which is parallel and nearest to the line segment oa. Suppose that a does not lie on one of the coordinate axes, or also that L is not parallel to one of these axes. Denote by Ll the segment intercepted on L by the axes. It has length 21b’(,where 6’ is the mid-point of L,.Now 21b’l > lal. Hence L , contains a point b of A . This point b has the property that {a, b} is a basis of A and that a, b lie in adjacent quadrants. There also exists such a point b if L is parallel to one of the axes. In what follows we denote by a‘ that one among the points +a, + b which lies in the second quadrant. Now let R be the (closed) cell of A spanned by a, b. Take a point y in r such that R’ = R +y contains 0. It may happen that o lies on the boundary of R’.But, by the hypothesis of the theorem, the vertices of R’ do not lie on the coordinate axes. Hence these vertices cannot all lie in two adjacent quadrants. Therefore, it is no loss of generality to suppose

556

INHOMOGENEOUS FORMS

CH.

7

that R’ has a vertex c, say, in the (interior of the) first quadrant and a vertex d in the third quadrant. It is clear that c - d # +a’ and that {a’, c - d } is a basis of A . We distinguish two possibilities. 1) a’ does not lie on the x,-axis. Then there are integers h, k 2 0 such that c -ha’, d+ (k 1)a’ lie in the region x , > 0 and c - ( h l)a’, d+ ka’ in the region x2 < 0. The points a’, d+ka’-(c-(h+l)a’) = d - c + (h+ k + 1)a’ form a basis of A , and the quadrangle with vertices c -(h+l)a’, c-ha’, d+ka’, d + ( k + l ) a ’ is a divided cell of r. 2) a‘ does not lie on the x2-axis. This possibility is dealt with in a similar way. This completes the proof of the theorem.

+

+

+

Theorem 2 is implicitly contained in the work of Segre [44a]. Later proofs of the theorem were given by Bambah [47a], RCdei [47a] and Suranyi [47a]. There is no analogue for grids in R 3 (Delone [48a], Birch [47a]). It is interesting to note that the proof of proposition 11) (which is a weaker form of theorem 2) cannot be based on the following argument. Let R be a closed quadrangle with vertices in r, one in each quadrant, and suppose that Rdoes not properly contain a quadrangle R, with vertices in r, one in each quadrant. Then R does not contain other points of r. In fact, the last conclusion is not justified in the case that R is not convex*. 47.3. We put the problem whether or not, for given Q and z, the inequality lQ(u-z)l (@(Q))* has infinitely many solutions. We shall see that,

under certain conditions, this is the case indeed. Since Q is indefinite, it is the product of two real homogeneous linear forms 4;, , 5 , , say. The determinant of these forms is equal to & (4b(Q))*, and so it does not vanish. For z E R2, we may write Q(x-2) = < , ( x - z ) ~ z ( ~ - z = )

(rl(x)-al)(4;2(x)-a2)7

with certain real numbers a, , a2. We prove Theorem 3. Let Q ( x ) = t1(x)t2(x). Suppose that t1 is not commensurable, i.e., not a multiple of a rational form. Then, for real a1 , a, and arbitrary

* See Mordell[42c]. Chalk [47a, 47b] dealt with the problem of determining infd(A), where the infimum is taken over the lattices A not having a primitive point inside the domain ((x,-l)x21 5 1. An application of the criticized argument leads here to a false result (see also Davenport [47a]).

§ 47 E

557

INDEFINITE BINARY QUADRATIC FORMS

> 0, one can solve

Proof. We may suppose that 4S(Q) = 1. Then the lattice A consisting of the points ( t , ( u ) , <,(u)), u E Y, has determinant 1. In virtue of our hypothesis, this lattice does not have points # o on the x,-axis. Therefore, by the results of sec. 45.4, it contains primitive points a = (al , a2) such that lal a,) I3 and la21 is arbitrarily large. For such a point a, let 52 and K be defined as in the proof of theorem 1. Then, as in that proof, A is a covering lattice of Q - l K . Moreover, if la,l is sufficiently large, Q-'K is contained in the region Ix1x2)< 3, lxll < E. From these remarks the theorem follows. Corollary. r f , in addition, the form rl(u)- u1 does not take the value 0 on Y, then ( 3 ) has infinitely many solutions.

Taking special forms Q we get Theorem 4. Let 9 be an irrational number. Then, f o r real u and arbitrary E > 0, there is a pair of integers u l , u2 such that

(4)

I(u1-9u2-a)u21

< 4, lu,-9u,-al

< E.

Corollary. If, in addition, u1 - 9u, - a does not take the value 0, then (4) holds for infinitely many pairs ( u l , u,) with u2 # 0.

Conversely, theorem 4 implies that ( 3 ) can be satisfied, at least with (iS(Q))' replaced by ($S(Q))*+q ( q > 0). For let tl(x) = X1-8x27 tz(x) = a Z l x l +a2,x2 and let al, a, be real numbers. Take a = a l . Let E be a positive number and let (ul , u,) be a pair of integers satisfying (4). Then M u ) - uz = az u1 +az2uz - uz =

hence

(azl 9 + aZZ)UZ

+ W),

CH. 7

INHOMOGENEOUS FORMS

558

Mordell [47d] proves theorem 3 by considering the three-dimensional domain (5)

d
3

xz)-azx31

1,

1x31

< 2,

where p is a positive number and t i ,t2 are linear forms of determinant 1. Actually, there are arbitrarily large values of p for which the domain ( 5 ) contains a lattice point u = (ul, uz , u s ) with u3 = 1. Further proofs of theorem 3 were given by Koksma [47a] and Jfijjsbjs [47a]. The corollary of theorem 3 is not generally true if the form t1( u )- a, does take the value 0. For let ~ l ( u o ) - a l = 0 and let u = u o + v . Then t,(u)-a1

=

<,@)> M u ) - % = t Z ( U ) + 0 ( 1 ) .

Applying the corresponding result for homogeneous forms we see that we can only assert that (3) has infinitely many solutions with ($3(Q))* replaced by any number > ($(Q))+ (Mordell [47e]). As to the corollary to theorem 4, HinEin [45d] showed that if 9 is irrational, a is any real number and E is any positive number, there are infinitely many pairs of integers u l , uz with I(ul -9u2 - a ) u21 < 5-*+.5, u2 > 0. Jogin [47a] proved this with E = O*. See also Koksma [DA], Kap VI and sec. 47.7. Theorem 3 is sharp in the sense that it ceases to be true if in the right hand member of the first inequality (3) we insert a factor 9 < 1. In fact, there are forms Q(x) = (,(x)tZ(x) such that the form is not commensurable and that p ( Q , Y ) ($6(Q))-* is arbitrarily near to 1. An example of such a form is Q(x) = (xi)' 2k x1x 2 - (x2)', where k is a large positive integer (see the relation (16) and the subsequent remark on the validity of the equality sign). Also, the constant 4in theorem 4 is best possible (Grace [47a]). The same is true with respect to the system I(u, - 9uz - a)u21 < 4, u2 # 0 (Kanagasabapathy [47a]).

+

47.4. Minkowski's estimate

p(Q, Y ) 5 (id(Q))* is not sharp if Q is not equivalent to a multiple of P. We deal with two general methods for

deriving sharper estimates. Of course, the new estimates will not be of the form p ( Q , Y ) S cp(S(Q)), with some function cp. First we deal with a method of Heinhold [47a]. For p > 0, let S, denote the star body I x l x z l 5 p. Then we have

*

See the review by K. Mahler (Math. Reviews 7, 274).

P 47

559

INDEFINITE BINARY QUADRATIC FORMS

Theorem 5. Let A be a lattice, with a basis A = {a', a'>. Suppose that S, contains the points +a', +a' and one of the points +(a1*a'). Then A is a covering lattice of S,. Proof. Replacing a2 by -a', if necessary, we see that it is no loss of generality to suppose that S, contains the points +a', +a2, +(a' +a'). For x , y E R2, let R(x, y ) denote the rectangle whose sides are parallel to the coordinate axes and have one end-point x or y . If y - x is one of the points a', a', a'+a2, then this rectangle is covered by the two domains x + S,, y + S,, on account of our assumption. Now the cell of A spanned by a', a' is covered by the five rectangles ~ ( oa'), , ~ ( oa'), ,

~ ( oa'+a'), ,

~ ( a 'a1+ , a2), R(a2, a'+aZ).

Hence this cell is covered by the domains x f S,, x theorem.

E

A. This proves the

Remark. If the conditions of the theorem hold and there exists a point z not lying in the interior of any body x+S,, x E A , then this point necessarily is congruent mod A to one of the points +a', +a', +(a'+a2). From theorem 5 we can easily deduce an estimate of the type meant above. Let Q be an indefinite binary quadratic form. Then Q = PA for some A . By theorem 5 , p(P,A Y ) 5 p if lP(+d)l S p , IP(+a')I d p and JP(+(a'+a')] p, for one of the signs. Hence, since p ( Q , Y ) = p(P,A Y ) and P(AAx) = Q(Ax) = A'Q(x) (A real, x E R'),

s

( 6 ) p(Q, Y ) 5

t max {IQ(L 0)L IQ(Oy 1)1, min (IQ(L1)1,lQ(L -1)l)).

This result was proved arithmetically by Barnes [47a] and Inkeri [47a]. The quantity p ( Q , Y ) is constant on each class of equivalent forms, but the right hand member of ( 6 ) is not. Therefore, for given Q, it is important to select a suitable equivalent form*. This is done by the following

Theorem 6. Let Q be an arbitrary indefinite binary quadratic form. For real P and let Q p , 0.1 Q p , u, 2 be defined by ~9

* Inkeri [47a] shows that there is an equivalent reduced form Q such that the right hand member of (6) does not increase if Q is replaced by Q.

560

CH. 7

INHOMOGENEOUS FORMS

Applying ( 6 ) to the forms Q p , u , l Qp,a,2 , one finds (9)

P ( Q ~ , ~Y. ) I ~max((p+c~)(2-lp--l), 1 - p 2 , 1 - g 2 ) if 0 2 p, o S 1

(10)

P ( Q ~ , Y~ ), 5~ $(l , +P)~

if 0 5 p 50 5 1.

These inequalities are an improvement of Minkowski's estimate, because ( N Q p ,

u,i>)'

( N Q p , u, 2))'

= 6(1 + P O ) = $(I+

2 +(P +

PI 2 HI +PI',

with equality only if the form in question is equivalent to a multiple of P. Proof of theorem 6. First suppose that at least one of the roots of the equation Q(<, 1) = 0 is rational. Then Q ( x ) is equivalent to a multiple of some form (xl -9,x,)x,, 0 3, 6 3. The last form is a multiple of Qp,u,l if we take p = 1, u = 1-209,. Next suppose that the roots of Q(<, 1) = 0 are both irrational. Then, by the results of sec.43.2, Q ( x ) is equivalent to a multiple of some form Q ( x ) = (x1-9,xZ)(xl-9,xz),whereO< 9,< 1,9,< -1, Q19,)=-1.

Case 1. 3, -9, (11)

> 2. Take p, cr so as to satisfy the equations (1+p)91 = 1-a,

(1-p)az = -(1+0).

Then Qp,,,, l(t,1) = 0 has roots 91, QZ and so Q is a multiple of Qp, u,1. Moreover, 0 S p < 1 and 0 5 0 < 1, on account of the inequalities o < 9, < 1, 9, < -1, 9192 2 -1, 91-32 > 2. Cast: 2.

(12)

Q1 -9,

S 2. In this case we solve p, cr from (1+p)9, = 1-0,

For these values of p and 0 < , p < 0 < 1.

0,

(1+p)9,

= -(l+a).

Q is a multiple of

Moreover,

Thus, in all cases, the assertion of the theorem holds true.

B 47

56 1

INDEFINITE BINARY QUADRATIC FORMS

A second method for estimating p(Q, Y ) was developed by Davenport [47b]. This method is of a purely arithmetical nature and makes use of the inhomogeneous minimum of a certain expression in one variable. The details are as follows. For z 2 0, let

+

q ( z ) = sup inf )(p k)'

(13)

- ri,

~k

where the infimum is taken over all integers k and the supremum over all real p. Actually, the supremum is attained (confer (46.9)). It is easily verified that

if 0 S z 5

(z-+)*

if z

if 4

+

and IpI S 3 5 z 5 3 and (27)*-1 6 p 5 (27)f 24 and l(z-+)*-pl S 3.

4-2

Thus p(z) 6 $-7, z, (z-i)* on the respective z-intervals just indicated. We now define a function $(z) by putting (14)

4%)

=

[

a 7

(z-+)*

f o r 0 5 r l i for + < z S t for z > 4.

This function $ ( r ) is non-decreasing, whereas p(z) 5 $(z) 2 0. We prove

for all

T

Theorem 7 . Let Q be arbitrary and let uo be a primitive point of Y with Q(uo) # 0. Then (15)

AQ,Y ) S

IaI$(a(Q)/(4a2)),

where a = Q(u0)-

Proof. It is no loss of generality to suppose that a = Q(uo) > 0 and that uo is the point (1,O). Then Q(x) = 4(x1)2+2Bx1x~+Y(x2)2, with certain real numbers p, y. We take an arbitrary point z = ( ~ 1 ,Z Z ) and substitute x = u-z, where u = ( u l , u 2 ) is a lattice point to be chosen below. We get

Q(x) = a{(ul -zi + P ( u 2 - ~ 2 ) / a ) ' -a(Q)((u2-~2)/a)~IThe last expression has the form a{(ul+p)' -z}, where p and z depend on Q and u2.We choose u2 such that lu2 -z21 54.Then 0S766(Q)/(4a2). Thereafter, we choose u1 such that I (ul + P ) ~-71 5 $(z). Then the theorem follows, because the function $ is non-decreasing.

562

INHOMOGENEOUS FORMS

CH.

7

In theorem 7, the point uo may be so chosen that la1 5 (6(Q))* or also S(Q)/(4a2) 2 $. For such a choice of uo, the right hand member of ( 1 5 ) is less than (+6(Q))*, because $(z) < J z for z 2 $. More precisely, uo may be so chosen that la1 is arbitrary near to the homogeneous minimum A(Q, Y ) . This leads to the following

Corollary. One always has p(Q, Y ) 5 ($d(Q)-$A(Q, Y)')*. Proof. If not Q w Q , , where Q , is the first Markov form, then A(Q, Y ) 5 (@(Q))%.The inequality for p(Q, Y ) then follows froril (15), with a replaced by A(Q, Y ) and with $(z) = (T-*)*. In the excluded case, the assertion can be verified directly.

+

We consider, in particular, the form Q ( x ) = (xl)' k x , x2- ( x 2 ) ' , k an integer 2 2. Applying theorem 7, or the corollary, we find

This result is exact if k is even. For then IQ(u,+$, u2+$)1 2 $k for all lattice points u = ( u l , u 2 ) (Davenport [47b]). But, if k >= 3 and k is odd, the inequality sign holds in (16) (Godunov [47a]). We conclude with some bibliographical remarks. Godunov [47a] gives a geometric proof of the corollary to theorem 7. HinEin [47a] obtains a similar improvement of the corollary to theorem 4. Barnes [47a] deduces theorem 7 from theorem 5. Barnes and Swinnerton-Dyer [47a] compute the function q ( z ) defined by (43), and Varnavides [47a] gives the best possible estimate q ( z ) 6 $(z) such that $(z) is non-decreasing. Perron [47b] and Eggan and Maier [47a] studied the quantity ~ ( y= ) sup {inf I(a-k)(b- k)l: a, fi real, Ia-PI

= 2y).

k

This quantity is easily seen to be equal to q ( y 2 ) . Inkeri [47b] and Cassels [50b, ~ O C ]find improvements of theorem 7. Heinhold [47a, 47b] treats the forms (x,-$x,)x,, where 9 = k/(2k+1) or lfk (k a positive integer), and Davenport [47b] investigates the Markov forms. One has

Y) = P(QZ Y ) = 3, ~ Q uY ,, Y ) < ( G A Q u , ')v = (i%-&m-'Y3 where m = det (el, UVe').

AQi

9

$9

3

8 47

563

INDEFINITE BINARY QUADRATIC FORMS

47.5. We come to the class of norm forms of real quadratic number fields. These forms have been investigated extensively. In the following, m is a

squarefree integer 2 2 and F, denotes the real quadratic field generated by J m . The algebraic integers in F, are given by the numbers 5 = u1 uzw, where u l , u2 are (rational) integers and where o is given by

+

(17) @

=

Jm [$(l + J m )

if m if m

=2 G

or 3 (mod4) 1 (mod 4).

The norm Nm 5 of such a number 5 is the value at the point u of the form N , given by

if m (18) Nm(x) = (xl)Z+xlxz-$(m-l)(xz)Z if m

{

(XA2

- m(x2>2

= (ul, uz)

=2

=

or 3 (mod4) 1 (mod4) (xER’).

The problem of determining the inhomogeneous minimum p(N,, Y ) is related to the question whether or not the euclidean algorithm holds in F,. We say that E.A. (= euclidean algorithm) holds in F, if to each pair of algebraic integers to,t1 in F, with tl # 0 one can find an algebraic integer q in F, such that (19)

INm (5o-r151)1 < INm 511.

If in (19) we divide through by INm tll, then the condition becomes: to each number p = z1+ z z o in F, (zl, z2 rationals) one can find an algebraic integer = u1 u i o such that

+

INm (p-q)l = INm ( u 1 - z 1 + ( u 2 - z z ) o ) I

=

IN,(u-z)l

< 1.

Thus we find that E.A. holds in F, if and only i f p ( N , , Y , z ) < 1for each rational point z. In particular, E.A. holds if p(N,, Y ) < 1. The following result has been obtained.

Theorem 8. E.A. holds in F, in the following 16 cases: (20)

m = 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73.

For the first 15 cases, this has been proved already some decades ago. For references see RCdei [47b]*. Here we indicate briefly some recent proofs.,

* RBdei asserted that E. A. also holds in F0,. Many years later this was disproved by Barnes and Swinnerton-Dyer [47a].

564

CH. 7

INHOMOGENEOUS FORMS

Heinhold [47a] infers from theorem 6 that p(N,, Y )

p k , where

t max (P,2q + 1 -P)

P:, =

I

if m = 2 or 3 (mod4), m = q2+p, 1 5 p 5 2q 4max (P,2q+2-P) if m = 1 (mod4), m = (2q+1)2+4p, 1 5 p 5 2q+1.

Using the remark to theorem 5 he then finds that E.A. holds for m = 2 , 3 , 5, 6, 7, 13, 17, 21, 33, 37,41; he gets inter alia the exact values P(N2 Y Y ) = P ( N 3 Y Y ) = 3, P ( N s Y ) = 4, P ( N 6 Y Y ) = 3. Y

See also Inkeri [47b] and Steuerwald [47a]. Barnes [47a] and Cassels [50b] treat the cases m = 11, 29, 57, 73. Varnavides [47b] disposes of all cases (20) by using Davenport's method treated in sec. 47.4. E.A. does not hold for other values of m,i.e., the list (20) is complete. The proof of this fact is rather difficult, and many mathematicians have contributed to the final result. References are given by Brauer [47a], Inkeri [47c], Cugiani [47a] and Barnes and Swinnerton-Dyer [47a]. A more recent and systematic treatment of the whole question is given by Ennola [47a]. See also sec. 48.3. We comment briefly other result for the forms N,. We observe that an expression such as N,(u-z) is defined for arbitrary z E R2, not only for rational points. Let m be fixed and let o be given by (17). Let z = (zl, z 2 ) be any point in R2 and let u = (ul ,u 2 ) be a point of Y. For a E F,, denote by a' the field conjugate of a. Further put p = z1+z20,

d

= z1+z20r,

5

= u,+u2o.

Then p and D are certain real numbers, whereas 5 runs through the algebraic integers in F, if u runs through the lattice Y. Now, by (18), N,(u-z) = ( u l - z 1 + ( u 2 - z 2 ) w ) ( u 1 - z ~ + ( u 2 - z 2 ) o ' ) = (t-p)(tr--d). Hence, (21) p(N,, Y, z )

=

inf {l(<-p)(t'-o)l:

5

analgebraicintegerinF,,,}.

Davenport [47c, 47d, 47e] investigates very carefully the expression in the right hand member of (21), in the case m = 5. He then finds a sequence of isolated minima of the inhomogeneous form N s ( x - z ) .

§ 47

INDEFINITE BINARY QUADRATIC FORMS

565

Varnavides [47c] obtains a similar result for N 2 . Inkeri [47d] proves that p ( N I 3 )= 3; he also shows that

l(t-p)(t’-c~)l > 3

for each algebraic integer 5 in F13,

if p = +w and CT = -$w, where w = +(l +J13). This means that the inhomogeneous minimum of N , , is not attained. A geometric method was used a.0. by Barnes and Swinnerton-Dyer [47a, 47b] and Bambah [47b, 47~1.Varnavides [47d, 47e7 and Godwin [47a] considered forms near to N , or N 5 . 47.6. We discuss a problem in inhomogeneous diophantine approximation. First we introduce a notation. Let 9 be a given irrational number. For real a, put

(22)

p(9, a) = lim inf J(u,-9u2-a)u2). IuzI-+m

By theorem 4, p(9, a) =< # if the linear form p1 - 9u2- a does not take the value 0. If this condition is not fulfilled, then p(9, a ) 2 5-’. Now let m be an integer 2 2 and let s, t be integers with g.c.d. (m, S, t ) = 1. We ask for which positive constants IC there are infinitely many fractions p / q satisfying (23)

19-p/q1 < xq-’,

p

= s (mod m), q = t (mod m).

The greatest lower bound of the numbers by ~ ( 9m, , s, t ) .

IC

with this property is denoted

The problem stated was attacked first by Scott [47a], in the case m = 2. In this case, the question is to find approximationsp/q such thatp and q are of given parities (not both even). Scott found that, for arbitrary 9, s and t, rc(9,2, s, t ) 5 1. Here the constant 1 is best possible. Koksma [47b] dealt with the general case. He found that ~ ( 9 m, , s, t ) B am”.

(24)

The proof of this result is a simple application of theorem 4. It runs as follows. Write p = p’m+s, q = q’m+ t . Then p’, q‘ are integers. Multiplying both members of the inequality (23) by q2 we get or also (25)

+

1(q’m t)(q’m9+ t9 -p’m-s)l < K,

I(q’+p)(q’S-p’-

a)I

< ~m-’,

566

INHOMOGENEOUS FORMS

CH. 1

where we have put p = t/m and a = (s- t9)lm. Now, since 9 is irrational and g.c.d. ( m , p , q ) = g.c.d. (m, s, t ) # my the form q f 9 - p f - a = = (qS-p)/m does not take the value 0. Hence, by theorem 4, there are infinitely many solutions of (25) if Icm-’ > 4.This proves (24). More precisely, the argument yields where a = (s-tS)/m. ~ ( 9 my , s, t ) = p ( 9 , c1), (26) The investigation was carried further a.0. by Kuipers and Meulenbeld [47a, 47b] and Descombes [47a, 47bI. The last author could easily show that the expression sup ( ~ ( 9m, , s, 2): 9 irrational) does not depend on s and t, provided g.c.d. (mys, t ) = 1. Denoting this expression by ~ ( mhe)proved among other things that (27) Ic(m) = (~~‘--~?1-1)(1-2/m)-~

if m is even and m > 2 if m is odd and m 2 13.

The main result of Kuipers and Meulenbeld is that ~ ( 3 = ) 3/45. The proofs of all these results make use of the theory of continued fractions. Tornheim [47a] proved that the system (23) has infinitely many solutions, with K = 5-:m2, if 9 is any real number and g.c.d. (m,s, t ) = l. 47.7. Some separate results are as follows. Chalk [47c] proves that each grid r = A + z has three points xl, x2, x 3 in the region Ix1x21 5 +d(A) such that (x2-x1, x3--x1) is a basis of A . Cassels [47b] extends Markov’s results (sec. 43) to inhomogeneous forms Q ( x - z ) ; among other things, he lists the forms Q ( x - z ) for which inf IQ(u -z)l L +(4S(Q)):, where the infimum is taken over the points u E Y with Q ( u - z ) # 0. K. Rogers [47a] generalizes the relation (6) to a class of forms F associated with star-shaped domains in R2.Bambah [47d] treats binary cubic forms F of positive discriminant; if such a form F is reduced in the sense of Hermite (see Davenport [42b]), then P ( K Y) 5 Q max (IF(1, O)l, IW, 1 ) I Y IF(L 1)1, IF(L - 1)1>. (28) Chalk [47d] proves that (28) holds for any binary cubic form F of negative discriminant. It should be observed that p o ( F ) = co if F is a binary cubic form of discriminant # 0, by Macbeath’s theorem 23.3 and the second relation (46.7); actually, the domain IF(x)l 5 1 satisfies the hypothesis of Macbeath’s theorem. Mordell [47f] generalizes Chalk’s result to a class of binary forms F satisfying certain general geometric conditions.

5 48

DELONE'S ALGORITHM. LOWER BOUNDS FOR

p(Q, Y )

567

Heinhold [47a] gives a formula for the inhomogeneous minimum of a positive definite binary quadratic form Q(x) = a(xI)' +2Bx1 x2 y(x2)'. If such a form is reduced, i.e., if 0 S B S a and a S y, then one has the exact result

+

To obtain this result Heinhold determines, for reduced Q, a number /i > 0 such that in the oval Q(x) = p there can be inscribed a hexagon H with vertices f a , k b , +c, where b = a+e' and c = a+e2; the plane is simply covered by the hexagons H + u , u E Y,and the number p is the required minimum p ( Q , Y ) . Further, Heinhold [47c] uses a similar method in the case of anarbitraryconvex distance function in twovariables. Finally, we discuss some results concerning the quantity ~ ~ (a) 9 =, lim inf I(ul -$u,-a)u,l,

(30)

YZ+W

where 9 is irrational and a is a real number. Here, in taking the limes inferior we account for pairs ( u l , u 2 ) with u2 > 0 only (compare (22)). By the result of HinEin [45d] mentioned already in sec. 47.3, we always have ~ ' ( 9 , a ) S 5-*. Cassels [50a] refines this result by showing that (31)

p'(9, a) S ~ ( ( ~ ~ + 8 ~ + 4 ) * - 1 c - l } , where

K =

p'(9,O).

Cole [47a] considers the case that u1 - 9u2 - a does not take the value 0; Cassels [47c] proves that in this case ~ ' ( 9 , a) 27/(28J7) and that this result is best possible. Cassels [47c] and Descombes [47c] give further results on the set of values of ~ ' ( 9 , a). See also theorem 48.5. In sec. 50 we treat asymmetric inequalities. -18. Delone's algorithm. Lower bounds for p ( Q , Y )

48.1. We continue the discussion of indefinite binary quadratic forms. First we deal with an algorithm of divided cells developed by Delone [48a]. It is related to a certain algorithm in the theory of diophantine approximation set up some decades ago by Morimoto*. The last algorithm is based on Klein's geometrical interpretation of continued

*

See Koksma

[DA], Kap. VI.

568

cn. 7

INHOMOGENEOUS FORMS

fractions and can be described arithmetically as the continued fraction expansion of a given number 9 to nearest integers. Delone's considerations are as follows (see also Barnes and Swinnerton-Dyer [48a]). Let r be a grid in R2 not having points on the coordinate axes. By theorem 47.2, there are divided cells of r. Let Po be such a cell and let a, b, c, d, be the vertices of r lying successively in the first, second, third and fourth quadrant. The sides ab, cd (which intersect the x2-axis) are called the x,-sides of Po and the sides bc, da are called the x,-sides. Suppose that the x,-sides are not parallel to the x,-axis, so that these sides produced intersect the x,-axis. Then there are integers h, k such that the pairs of points (1)

a-h(d-a), a-(h+l)(d-a)

and c + k ( b - c ) , c + ( k + l ) ( b - c )

both straddle the x,-axis. The integers 12, k are both positive or both negative, and the four points (1) are the vertices of a new divided cell PI,say. The process described can be repeated if the x,-sides of P, are not parallel to the x,-axis. It can also be inverted: if we produce the x,-sides of P, and take on them pairs of points of r straddling the x,-axis, then we get back Po.In this way we get a certain chain of divided cells P, (r 5 0) of I'. Each pair of consecutive cells P,, P,+l in this chain has the property that the vertices of P,+ lie on the sides produced of P,,and the vertices of P, on the x,-sides produced of P , + l . If Pi has vertical x,-sides for some index r' 2 0 and also if P,..has horizontal x,-sides for some index r" S 0, then the chain terminates there. Each of these two possibilities may or may not arise. Thus the chain obtained is a finite chain, a one-side infinite chain or a two-sided infinite chain. It certainly terminates, on the right or on the left, if there are horizontal or vertical vectors with end-points in r. For then we can easily find divided cells with horizontal or vertical sides, respectively, and these cells occur in the chain, in virtue of the theorem to be proved below. For arbitrary r, let us denote the length of the x,-sides of P, by p,, the length of the x,-sides by up, the horizontal breadth of the P, (i.e. the breadth of the thinnest vertical strip containing P,) by p, and the vertical breadth of P, by 7,. With these notations, we can state

,

Theorem 1. Let r be a grid in R2 not having points on the coordinate axes and let {P,} be a (maximal) chain of divided cells of r. Then the following properties hold.

0 48

DELONE'S ALGORITHM. LOWER BOUNDS FOR

p(Q, Y)

569

1". If the chain {P,} continues indejinitely to the right, then pr + co and p, + 0 as r -+ co; fi the chain continues indejinitely to the left. then o r + co and y, -+ 0 as r + -a. 2". Each divided cell of I' occurs in the chain. 3". If r is admissible for S: Ix1x21 5 1 and if x is a point of r with (xix2(c J, then x is a vertex of some divided cell of r.

Proof. 1". Suppose the chain continues indefinitely to the right. Take a straight line L orthogonal to the x,-sides of Po and, for r 2 0, project the xi-sides and the x,-sides of P, onto L ; let the projections have lengths p i , o:, respectively. From the construction of P, it follows that, for r 2 1, (2)

0:

=

PL-1,

P: = trp:-,+o:-,,

where the t, are integers 2 2. Moreover, ob = 0, by the choice of L. It follows that p:/pb is integral and strictly increasing for r = 1,2, . . Hence, p i increases indefinitely for r + 00. The same is true for p r . Since the cells P, all have area d ( T ) and intercept intervals of increasing lengths on the x,-axis, it follows that 8, + 0 as r -, n3. In a similar way the assertions for 0,and y r follow.

..

2". Suppose the chain {P,}continues indefinitely in both directions. Let R be an arbitrary divided cell of I'. Let a, b, c, d be the vertices of R ( a in the first quadrant, and so on) and denote by L', L - , M ' , M the straight lines containing successively the sides ab, cd, ad, be. For arbitrary r, let L: , Lr-, M: , M i be defined analogouslyfor P,.Finally, let z be the point where L- intersects the x,-axis and, for arbitrary r, let Z' be the point where Lr- intersects the x,-axis. The x,-coordinates of the points 'Z are strictly decreasing and tend to 0 for r + -co and to - 00 for r + 03. This is easily deduced from property 1 ". Consequently, we can choose the index r such that either z = z' or z lies strictly between z' and 2'". We prove that, with this choice of r, necessarily z = zr and

R

= P,.

Let a', b', c', d' be the vertices of P, and let y , y' be the points where M,., M:, respectively, intersect the x,-axis. Since y, y' are not points of I' and zlies strictly between Mr-, M:, the triangle yy'z does not contain points of I'. Therefore, the line L - - which passes through a point of I' in the third quadrant and also through a point of in the fourth quadrant - does not pass between the points y, y'. Hence the points a', b' lie strictly above L-. If c' should also lie strictly above L-, then a', c' should lie on

570

INHOMOGENEOUS FORMS

CH.

1

or above L + , which is impossible. Similarly, it is not possible that d' lies strictly above L-. Hence, by the choice of r, then points c', d' both lie on L- (so that z = z") and so L- = Lr-. The last fact entails that Lf =L,!. It is now clear that R coincides with P,. The proof just given is easily adapted to the case that the chain {P,} terminates, on the right or on the left. 3". It is no loss of generality to suppose that x lies in the first quadrant and that xl = x2. Then, by our assumption, x1 = x2 < 43. For convenience's sake, we also assume that the chain {Pr> is infinite ih both directions. Now let L be the straight line y , + y 2 = x1 + x 2 = 2x1 and let G be the segment of the domain y l y 2 2 1 cut off by L. Intersecting an arbitrary straight line containing x with the hyperbolae yly2 = & 1 and using the relation x1 = x2 < Jfwe see that the segment G has the property that 2G-x is contained in G v int s. Since r is assumed to be admissible for S , it follows that has no point # x in G. So the triangle T bounded by L and the coordinate axes does not contain a point # x of r. Our next step in the proof of 3" is the choice of a cell P,. For arbitrary r, let the straight lines L:, Lr-, M,', Mr- be defined as above. The lines L,? ,Lr- tend to the x,-axis for r -+ - co and the lines M,', Mr- tend to the x,-axis for r + 03. Moreover, for all Y, the lines LT, Lz are the same as the lines M:- 1 , Mr-- (taken in some order which need not be the same for all r ) . It follows that there exists an index r such L,!, Lr- make an angle 5 45" with the positive x,-axis and M:, M,- make an angle 45" with the positive x,-axis. Since the line ox is the bisectrix of the first quadrant, we may conclude that x lies strictly above L,- and strictly to the right of Mr-. Hence, since x is a point of this point lies on or above L: and on or to the right of M:. The point of intersection of L:, M: then lies in the triangle T.Hence, by what we have proved already, this point coincides with x. This proves the assertion 3".

r

r,

In what follows we denote the vertices of the original cell Po of r by uo, bo, co, do instead of a, b, c, d, so that u0 lies in the first quadrant, and so on. More generally, for arbitrary r, we denote the vertices of P, by d, b', ,'c d'. The notation is so chosen that for all r for which P, is not the last cell in the chain, we have relations of the type (3)

a'+' (cr+l

- ar-(h,+l)(ar-d'),

1c'+(k,+l)(b'-c'),

b'+l = d - h r ( d - d ' ) d r f l = Cr+kr(br-c').

5 48

DELONE'S ALGORITHM. LOWER BOUNDS FOR p(Q,

571

Y)

Here h,, k, are integers with h,k, > 0. Observe that a'-d' = b'-c'. It should be stressed that, with the notation chosen, it need not be true that, for all r, a' lies in the first quadrant, b' in the second quadrant and so on. Actually, a"' and b'+' lie in the same quadrants as a' and b' if h, < 0, and in the opposite quadrants if h, > 0. Thus, for all r, the points a', b', cr, d' either lie successively in the first, second, third and fourth quadrant or in the third, fourth, first and second quadrant. For a given grid r not having points on the coordinate axes and any given Po the relations (3) define a certain sequence of pairs of integers h,, k , with h,k, > 0. We agree to put

h,

(4)

=

k,,

= CO,

h,--l

=

k,.*-, = co

if the chain {P,]terminates on the right with P,. or on the left with P,.,, respectively. Further, an arbitrary chain of pairs of integers h,, k,

.(

with h,k, > 0 is called exceptional* if there is an index ro such that one of the following possibilities arises:

(5)

h, = -1 = -1 hro+2,= k,o+2r+l = 1

for all r 2 ro or for all r for all r 2 ro or for all r for all r 2 0 or for all r

ro ro 0.

In each of the 6 cases considered the chain {(h,, k,)} is supposed to continue indefinitely in the direction in question. With the help of the notion of an exceptional chain {(h,, k,)} a result of Delone can be expressed as follows: Theorem 2. If I' is a grid not having points on the axes, ihen any associated chain { (h,, k,)} is not exceptional. Conversely, i f any chain of pairs of integers h,, k, such that h,k, > 0 for all r, is not exceptional, then this chain is associated with some grid r which does not have points on the axes. Proof. Let r be a grid not having points on the axes and let (P,}be a (complete) chain of divided cells of r. Consider the associated chain {(h,, k,)}. Two cells P,, P,+l have a vertex in common if and only if h, = k 1 or k, = 1; more precisely, we have

* We 'always assume, without mentioning it explicitly, that in either direction the chain continues indefinitely or ends with a pair (w, w). Thus a finite chain is not exceptional.

512

CH.

INHOMOGENEOUS FORMS

b'" ,r

= d' if h, = 1, + 1 = u" if h, = -1,

drfl = br if k,

cr+' = cr

=

7

1,

if k, = -1,

on account of (3). Just if one of the possibilities (5) arises, for some index ro ,the cells P, have a common vertex from the index ro onwards or downwards. But such a common vertex necessarily lies on one of the axes, in contradiction with our assumption. Hence the chain {(h,, k , ) ) is not exceptional. Now suppose we are given an arbitrary chain {(A,, k,)) which is not exceptional and has the property that Arkr > 0 for all r. We construct a chain of parallelograms as follows. Take an arbitrary parallelogram Po in the plane (at the end of the proof, Po will be subjected to a suitable affine transformation). Denote the vertices of Po by a', boycoydo in such an order that aobo, boco, codo and doao are the sides of P o , and define a chain of parallelograms P, with vertices a', b', cry d' by applying the formulae (3), forwards and backwards. In an obvious sense, we may speak of the x,-sides and the x,-sides of the P,. Suppose the chain {(h, ,k,)} continues indefinitely to the right. Then the x,-sides of any P, are intersected by the x,-sides of P,+ and the intercepted intervals are at most half as long. Therefore, for r + 00, the x,-sides of P, tend to a uniquely determined limit line M , say, which intersects the x,-sides of all P,. It is even true that M intersects these sides at inner points, because the chain {(A,, k,)} is not exceptional. If the chain {(h,, k,)} terminates on the right, then too there exists a line M with this property, although it is not uniquely determined in this case. Likewise, there exists a straight line L intersecting the x,-sides of all P, at inner points. Now perform an affine transformation which maps L and M onto the xl-axis and the x2axis, respectively. Then the grid r generated by the vertices of the image of Po under this transformation has the required property. In particular, let us consider the chain {( + k , k k ) } ,where pairs (k, k) alternate with pairs (- k, -k ) and where k is a fixed integer 2 2. By theorem 2, this chain is associated with some grid r,; moreover, this grid is determined uniquely up to linear transformations leaving invariant the axes, i.e. transformations of the type x i = ax1 ,x i = pxz (a,p # 0). On account of the symmetries of the chain { ( + k , + k ) } , that transformation of the above-mentioned type which sends a given vertex of a given divided cell of r k into any other such vertex, is an automorphism of r k ,

I 48

DELONE'S ALGORITHM. LOWER BOUNDS FOR

p(Q, Y )

573

and therefore it has determinant k 1. Hence Ixl x21 has the same value for all vertices of all divided cells of r k . We may suppose that this value is 1 and that there is a divided cell of r k whose x,-sides fall along two lines x1 +x, = &A (A > 0). The line x1 +x, = ,I intersects the hyperbolae xl xz = & 1 in four points all belonging to rk,and the interval intercepted on it by the axes contains just k points of r k . A simple computation now yields or also

22 d(rk) = __ (1'-4)* k-1

'A

=

+

2( k2 1) , k ~

= 4(1+ k-')'.

Clearly, d(rk)--f d'(S) = 4 if k + co. An inhomogeneous indefinite binary quadratic form corresponding with the grid r k is given by Q ( x - z ) , where Q(x) = ( 2 / k ) ( ( ~ , ) ~ + 2 k x ~ x ~and - ( xz~=) ~(+, ) The form Q has inhomogeneous minimum 1. Thus we have refound certain results of Kanagasabapathy [47a] and Davenport [47a] (see the end of sec. 47.3 and the relation (47.16)). Inspecting the grid r3one finds that the constant $ in theorem 1 is best possible.

e).

48.2. We derive some analytical formulae. We suppose that the chain

{Pr>continues indefinitely in both directions. For arbitrary r, put (6)

= hr+kr,

m, = . h , - k , ,

zr = a r - d r =

V-c'.

Then, by (3),

+1

= -l,zr+(dr-cr)

(7)

zr

(8)

d + c r = mrzr+

--

= -lrzr+zr-'

+ms~,zS-'+us+cS

( r < s).

The first formula yields (9)

+

zs = (- l y - r ( p r , ,zr q,, zr- ')

( r , s arbitrary),

where P,,~, q,,, are integers satisfying the recurrence relations (10)

Pr,s+l

=

ls+1Pr,s-Pr,s-19

qr,s+l

= ls+l-qr,s-l-

Barnes and Swinnerton-Dyer [48a] apply the formulae (8) and (9) to derive an expression for the inhomogeneous minimum p(S,r ) only in-

CH.

INHOMOGENEOUS FORMS

514

7

volving the determinant d ( T ) and the numbers l,, m,; they use that { p ( S , T ) } 2= inf{Ix,x21: x a vertex of a divided cell P,},

zl,, -, 0 for r + co,

z2,' -, 0 for r

+

co

(z' = a'-a').

Barnes [48a, 48b J uses the result to compute the inhomogeneous minima of the norm forms N , , and N46 and to prove the following

Theorem 3. Let p be a real number with 0 5 p 5 t. Then there are continuously many irrational numbers 9 to each of which one can.find continuously many real numbers u such that p(9, u ) = p. Here p(9, u ) is the expression given by (47.22). 48.3. We now come to the problem of deriving lower bounds for the

inhomogeneous minimum of an arbitrary indefinite binary quadratic form Q. Davenport [48a] proved that there exists a universal positive constant K such that p(Q, Y ) > 7c(46(Q))*

for each Q .

On account of (46.18) this means that the lower absolute inhomogeneous minimum p o ( P ) of the form P ( x ) = x1x2 is positive, or also that p o ( S )is positive (so that S has a finite covering constant), where S is the star body lxlxzl 5 1. In sec. 47.1 we found already that p o ( S ) = 3. Thus the star body S has the property that its absolute inhomogeneous minima both are positive and finite. It may be expected that, in general, two-dimensional star bodies with this property are of a similar shape as S. Later on, Davenport [48b] proved that (11) holds with K = & and that, if Q is a rational form, there exists a rational point z with p ( Q , Y , z ) > &(46(Q))*.Animportantconsequenceisthatp(Q, Y,z)>,l, for some rational point z, if Q is a rational form with 4S(Q) 2 128'. Hence, E.A. does not hold in the quadratic field F, if 46(N,) 2 1282. Thus there are onlyJinitely manyJieldsF, in which E.A. holds (see sec. 47.5). Davenport's proofs use chains of reduced forms and are related to the method of Morimoto referred to above. A geometric way of looking at the question is as follows. Take a grid r not having points on the axes and consider the chain of divided cells P,.The x,-sides of any P,intersect, by definition, the x,-axis and have end-points a,, d', or b', c', which lie nearer and nearer to the x,-axis if r -+ - co. Similarly, the x2-sides of P,

0 48

DELONE'S ALGORITHM. LOWER BOUNDS FOR

p(Q, Y)

515

intersect the x,-axis and their end-points approach the x,-axis for r + co. The idea is now to shift the grid r (thereby changing the chain of pairs of integers h,, k,), in a horizontal and also in a vertical direction, such that the x,-axis intersects the x,-sides of the P, with r 5 0 in points which are as near as possible to the mid-points of these sides, and that the x,-axisintersectsthex,-sides of the P, with r 5 0 in points near to the mid-points of the latter sides. A method of Cassels [48a] closely resembles the procedure just sketched, although it does not use properties of divided cells. Cassels proves

A,

A.

Theorem 4. The relation (1 1) holds with IC = so that we have p o ( P ) 2 Moreover, if Q is a rational form, then p ( Q , Y, z ) > (4S(Q))* for some rational point z.

We sketch the proof of the first assertion of this theorem. First we deal with the case that Q is a non-zero form, i.e., that Q(u) # 0 for each point u # o of Y. Then Q may be assumed to be a form of the type Q(x) = (xi -9xz)(xi -VXZ), (12) where 9 and cp are irrational numbers. Let p = (4S(Q))* and let A be the lattice of points ( u 1 - 9 u , , u,-cpu,). Then d ( A ) = I9-pl = p. Further, let a be a real number > 2 to be chosen later. The proof now consists of the following three steps.

I. There exists a two-sided injinite chain of points a'

=

( a l , , , a,,,)

in A such that (13)

Ia1,rI 5 O-'Ial,r-lI;

Ia2,rI

5 CpIal,r-lI- 1 ;

Ia2,rI

(r (14) a l , , , a 2 , - , + O

for r

+ 00;

5

Iaz,r-lI

= 0,

+1,

I a l , - J , laz,,I + co for r

. . .)

+

co.

A finite chain having the properties (13) can be found by applying the theorem of Minkowski to a sequence of suitably chosen quadrangles with sides parallel to the coordinate axes; a suitable diagonal process then leads to an infinite chain of points a' E A having the properties (13) and (14). 11. There is apoint x o E R2 such that (15)

ldet (xo-a, a')! 2

a-2

___ p

2(a- 1)

f o r all r and all points a E A.

516

INHOMOGENEOUS FORMS

CH.

1

In fact, taking an arbitrary positive index rn and considering the sequence of numbers al,,,, al,"'- , . . ., ul, -,,, (which are increasing in absolute value on account of (13)), one can easily find a real number y2,"',say, such that 0-2

for r = m, m - 1 , .

. ., --m.

Here ]]all means the distance of tl to the nearest integer. Since det (a, a r ) is a multiple of d(A) = p for all Y and all points U E A , the point y"' = (0, y 2 , , , )then satisfies the relation

.")I 2

Idet(y"'-a,

0-2 p 2(0 - 1)

for a E A and Irl 5 m .

~

For m = 1,2, . . . let x"' be a point in a given fundamental cell of A with xm = y" (mod A ) . Let x o be a point of accumulation of the points x"'. Then xo satisfies the assertion. 111. There exists a positive constant IP(xO-a)l 2

(16)

only depending on

K:

t~

such that

for azz U E A .

Icp

For let a be a point of A and let y = xo--a. Then y1 # 0 and y 2 # 0, on account of (15) and the first relation (14). Put w = ( ~ p / ( y , y ~and ))~ choose an index r such that la2,rl

5 OIYZI

5

laz,r+ll.

Such an index exists on account of (14). By (13), we have Ia1,rI

5 0PIaz,r+lI- 5 1

~ P ( ~ I Y Z I ) -= ~

WIYlI.

Also, lal,ra2,rl5 p. Thus we have lyza1,rl

5 ~ I Y ~ Y Z I ,lYla2,rl S

WlYlYzl,

lY2a1,rl * Iyla2,rl

5 PlYlYZl.

Consequently, IdeG, ar)I S IY1a2,rI+IY2a1,rI 5 6 WlY,Y2l+PlW = (PIY1Yz1)3 Using (15) we ultimately get IYlYZl 2

KPY

The last expression is greater than

for 0 = 5.5.

-

(0++0-*).

0 48

DELONE'S ALGORITHM. LOWER BOUNDS FOR

517

p(Q, Y)

Next we consider the case that Q is a zero-form. Take a sequence of non-zero forms Q, and a sequence of points z"' such that Q, + Q for m + co and that p(Q,, Y , z") = p(Q,, Y )for all m. We may suppose that the points zm lie in a bounded region and even that they converge to some point z. Then we have p(Q, Y) 2 P(Q, Y, Z) 2 'im

SUP

m-r m

P(Qm >

2 IC lim (46(Q,))* m-rm

X

z") = lim SUP (Qm Y) 2 3

m-r m

= IC(~S(Q))*,

in virtue of theorem 46.2 and the result just obtained. The second assertion of theorem 4 does not follow trivially from the first one; the proof of it requires the construction of a sequence {ar>with certain periodicity properties. Cassels also shows that p(Q, Y , z ) > &(4S(Q))+ for continuously many points z and that p(Q, Y, z ) = 0 for almost all z, the form Q being fixed arbitrarily. Ennola [47a] proves that (11) always holds with K = Pitnam [48a] proves that p(Q, Y ) &(48(Q))* = (16+6J6)-l > for the class of Markov forms; she uses Delone's algorithm.

=-

A.

48.4. We may apply theorem 4 to any form Q(x) = (xl - 9x2)x2,where

9 is a given real number (rational or irrational). We find that there exists a point z such that lQ(u-z)l = I ( u 1 - ~ u 2 - ( z 1 - 9 z 2 ) ) ( u 2 - z 2 ) ~> &

foralluE Y.

Clearly, it is no loss of generality to suppose that 0 S z2 < 1. Then Iu2-z21 S lu21 if u2 > 0. So we have Theorem 5. There exists a positive constant K with the following property: for each real number 9 there exists a real number u such that (17)

I (ul - 9u2 - u)u21 > K for all pairs

( u l , u2) with u2

> 0.

For K one may take the constant A. Improvements of this theorem were given by Davenport [48c], Prasad [48a] and Godwin [48a]. The latter found that 68/483 = 0.14078.. is an admissible value of IC,but that 0.21 14 no longer is. For older results see Koksma [DA].

.

578

INHOMOGENEOUS FORMS

CH.

7

49. Inhomogeneous forms in more variables 49.1. First we say a few words on inhomogeneous minima of positive definite quadratic forms. Let Q be a given positive definite quadratic form in n variables. Then Q can be written as QLo', where

(1)

Q"'(x)

=

]XI'

= (x~)'+

* *

-

+(A$

is the square of the distance function of the unit sphere K,, and where A is some non-singular linear transformation of R" (see sec.,38.1). The form Q has discriminant D = ldet A)'. By (46.12) and (46.17), (2)

p(QYY ) = P(Q('), A Y ) = {p(K,,, AY)}'.

Thus the problem is what we can say about the quantity p(K,,, A ) , as a function of A . The sphere K,, is a bounded convex body. So its covering constant T(K,)is positive and finite, whereas its inhomogeneous determinant A'&) vanishes (secs. 21.1 and 23.2). Hence, by (46.7), po(K,,)is positive and finite, and po(K,,) is infinite. Using (46.7), (2) and the relation D = (det A)' we obtain Theorem 1. The expression p(Q, Y)D-l'", Q a positive definite n-ary quadratic form, is unbounded and has greatest lower bound pO(Q")) = = {PO(Kfl)}' = {Wfl)l-"". The lower bound is attained if Q = QLo) and AY is homothetic to a maximal covering lattice of K,, (see theorem 21.3). For T(K,,) we have the formula (3)

T ( K ) = K?li%, where K,, is the volume of K,, and 9, = 9(KJ is the density of thinnest lattice covering of K,,. Several authors have given lower or upper estimates for 9;, these estimates correspond with lower and upper estimates, respectively, for po(K,,). In sec. 21 we discussed already known upper bounds for 9(K), where K is an arbitrary bounded o-symmetric convex body, the strongest result being given by (21.24). For 9,,, Rogers [21d] gave the slightly sharper estimate 9, = qn (log n)(log 2 4 l l o i 4) as n + co. (4)

I 49

579

INHOMOGENEOUS FORMS IN MORE VARIABELS

Older results were given by Davenport [49a, 49bl. In the first paper cited Davenport showed that, if m is a positive integer and pm = = ((n/12)(1+2rn-2))3, the special lattice of points x = (xl, ., x,) with

..

xi = ui+u,/m

(i = 1, . . ., n- I),

x, = u,/m

(ul, . . ., u, integers)

is a covering lattice for p m K , . Hence T ( K , ) 1 l / ( m p m " ) or also 9, 5 rC,mp,"; taking m = [(2n)*] one gets (5)

9, c GI,,",

where GI,

-+

( 7 ~ 4 6 ) for ~ n

-, co.

On the other hand, it is trivial that

(6)

9, 2 1

for all n.

A non-trivial lower bound for 9, was given by Bambah and Davenport [49a] who showed that 9, > +-q,, where E, -, 0 for n -, 03. This proof uses the following idea. Let A be a covering lattice of K, and let S be the set of points for which o is the nearest point of A . Thus S is the parallelohedron of Voronoi determined by the lattice A and the ordinary distance function 1x1. Then S is contained in K,, because A is a covering lattice of K,; moreover, the parallelohedra S+x (x E A ) cover simply the space R".The authors now estimate the number of faces of S and then derive an upper bound for the volume of a convex polyhedron contained in K, which has a given number of faces. Erdos and Rogers [49a] derive a similar, but weaker estimate for 9 : = 9*(K,). A larger lower bound for 8,*, and so for 9,,has been obtained by Coxeter, Few and Rogers [49a]. They use again 'Voronoi polytopes'; dissecting these polytopes into simplices and using similar arguments as in the proof of Rogers's theorem 38.3 (on packing of translates of K,) they find that (7)

Returning to quadratic forms we may say that, by (4) and (6),

(8)

p 0 ( ~ ( O )= ) (K,/Q-~/~

- -

IC;"~ n/(2ze)

as n

-+

co.

This formula may be compared with the relation (38.20) for Hermite's constant 7, = A(Q(").

580

CH.

INHOMOGENEOUS FORMS

1

For n = 2 and n = 3 the exact value of 9, is known. One has 9, = 2n/(3J3),

(9)

93 = (5nJ5)/24.

The first relation (9) follows from theorem 22.5; the second one was proved by Bambah [49a], Barnes [49a] and Few [49a]. The proof by Few uses only elementary geometry. A maximal covering lattice of K3 is the 'body-centered' cubic lattice generated by the vertices of a cube with centre at o and with side length 4/J5 (so that r ( K 3 )= 32/(5./5)). Bambah [49b] proves that 4n2/(15J3) 5 Q4 S 2n2/(5J5) and he conjectures that Q4 = 2n2/(5J5). 49.2. Next we consider products of n inhomogeneous linear forms in n variables. Let S denote the star body IP(x)l j 1, where P ( x ) = xlxz* * x,,

and let A = ( a i j ) be an arbitrary non-singular linear transformation of R".Then P A ( x ) is the product of the n real homogeneous linear forms (10)

&(x)

+

= ail x1

- - + ainx,

(i = 1,

. . ., n).

A conjecture usually attributed* to Minkowski asserts:

1) d'(S) (11)

=

2" or also p o ( P ) = 2-", so that

p(P,, Y , z)

2-"(det A(

f o r all A and all z;

2) the equality sign holds in ( 1 1) if and only qf P A z P and z (mod Y).

= (4,.. ., i)

Minkowski himself proved the conjecture to be true for n = 2 (see sec. 47.1). Later on, it was proved for n = 3 by Remak [49a] and for n = 4 by Dyson [49a]. The proofs are complicated. Remak works with a dissection of the space R 3 into parallelohedra (honeycombs) and with reduced positive definite ternary quadratic forms. His proof was simplified by Davenport [49c]. Geometrically speaking, Davenport's proof consists in the deduction of the following two propositions.

I)

If A

is a lattice in R 3 , then there exists an ellipsoid E u ( y l ) ' + + j ( y 2 ) 2 + y ( y 3 ) 25 1 such that A is admissible for E and has three independent points on the boundary of E.

*

See Koksma [DA], p. 18.

8 49

58 1

INHOMOGENEOUS FORMS IN MORE VARIABELS

11) I f A has determinant 1 and the ellipsoid E is a sphere (so that the successive minima of A with respect to the unit sphere Kn are equal), then each sphere in R 3 of radius ($)* contains apoint of A .

From I) it follows that any given form P A , in the case n = 3 , is a multiple of some form P, such that the conditions of 11)hold for A = B E 11) implies that for all z E R 3 there exists a point y E A such that ((YI - Z i ) ( Y z - z z ) ( y 3 - z 3 ) l

5 5~

I9.

3 ( ~ ~ , - ~ l ~ z + ~ ~ z - ~ z ~ z + ~ Y 3 - ~ 3 ~ 2 )

-=

The case that the inequality I(yl -z1)(y2-z,)(y3-z3)~ Q does not hold for any point y E A is easily analyzed. Dyson derives his result by deducing four-dimensional analogues of I) and II), with the constant ($)* replaced by 1. The proof of the analogue of I) requires tools from algebraic topology. Mahler [49a] proves 11) arithmetically, under weaker conditions. Davenport [49d], Clarke [49a] and Samet [49a, 49b] give detailed results for special cubic forms P A . Varnavides [49a] derives an isolation theorem. An estimate for p o ( P ) valid for arbitrary n, was found by Cebotarev [49a]. His proof is slightly simplified if use is made of theorem 26.6 which, under certain conditions, assures the existence of a critical grid. Cebotarev's result is given by

Theorem 2. For all n

1 2, d'(S) 2 2)" or also p o ( P ) 2-*".

Proof. The star body S is automorphic. Each straight line passes through the interior of S, because the coordinate hyperplanes belong to int S. Further, there are S-admissible grids. For if A = AY is an S-admissible lattice, then the points Au E A with u1 * * +u, = l(mod 2 ) form a sub-grid of A not containing the point 0.This sub-grid is admissible for S. From theorem 26.6 and the remark following the proof of that theorem we now infer that there exists an S-critical grid r, say, which has a point z on the boundary of S. It is no loss of generality to suppose that z is the point (1, 1, . . ., 1). We show that F has determinant 1 2*". Let A be the corresponding lattice, so that I' = A + z . Then

+

n

IP(z+x)J =

i=l

11 +xil 2 1

for each point x = (xl ,

. . .,x,) E A.

582

Hence, since A is symmetric with respect to n

(12)

CH.

INHOMOGENEOUS FORMS

n 11 - ( x ~ ) ~ I 2 1

1

0,

for each point x of A .

i= 1

The last inequality does not hold for any point x # o with lxil < J2 ( i = 1 , . . ., n). Thus A is admissible for the cube K lxil < J 2 (i = 1, ..., n). Hence, by Minkowski’s theorem, d ( T ) = d ( A ) 1 2fn. Since r is an S-critical grid, this proves the theorem. Geometrically speaking, the proof just given is based on the fact that the union of the two domains S k z contains the cube K to which then the theorem of Minkowski is applied. We can derive a sharper result by applying Blichfeldt’s theorem to a suitably chosen set T with BT c c ( S + z ) u (S-z). It is easily verified that the last relation is fulfilled if we take ( i = l , ..., n ) ) .

T = { x : l x i l S l ( i = l , ..., n ) o r I x i - 1 1 5 1 - + J 2

So we have (Mordell [49a]) (13)

d’(S) 2 2f”(l+(J2-1)”).

Davenport [49e] applies Minkowski’s theorem to a convex body K , c ( S + z ) u ( S - z ) of a more complicated structure. He finds that (14)

d’(S) 2 2%4,,

where

c1, + 2e-

1 for n

+ 00.

A new idea was introduced by Woods [49a]. He proves the following lemma which is a refinement of Blichfeldt’s theorem.

Lemma 1. Let A be a lattice in R”,let y be an arbitrary point and let T be a bounded Jordan measurable set not containing two distinct points a, b with a-b E A. Then (15)

d(A) 2 2 V ( T ) - V(T’),

where Ty = ( T + y ) n (

u (T+x)).

XEA

Proof. Put T‘ = (T+y)\UXeA(T+x).

Then T u T‘ does not contain two distinct points a, b with a-b E A. Hence, by theorem 16.2, d ( A ) 2 2 V ( T u T’). Since V ( T u T ’ ) = V ( T ) +V ( T ’ ) = 2 V ( T ) - V(T,), this proves the lemma. The lemma can be applied as follows. Let

r = A+z

be chosen as in

5 49

583

INHOMOGENEOUS FORMS IN MORE VARIABELS

.

the proof of theorem 2, with z = (1, . ., l), and let T be the cube Jxil 5 2-* (i = 1 , . . ., n). Then the following assertions hold. 1) 2) 3) 4)

The cubes T + x , x E A , have no inner point in common. If T + z meets int T + x , then int T + x contains a vertex of T + z . If y is a point with IP(y)l 2 1, then V ( T n ( T + y ) ) i(42-1)". IP(y)I 2 1 for each point y ~ A + z .

By 1) and 2), T + z overlaps at most 2" cubes T + x , x 3) and 4), V ( T z )S 2"(J2- 1)" and so

E A.

Hence, by

2 V ( T ) - V ( T z ) 2 2*"(2-(2-J2)"). Hence, by lemma 1 and the choice of (16)

d'(S)

r,

2 2f"(2-(2-J2)").

By combining the ideas of Davenport and Woods, Bombieri [49a, 49b] and Gruber [49a] both proved that d'(S) 2 3 2%,,, where a,, 4 2e- 1 for n 4 co. The first author made use of Parseval's formula (confer Siegel's proof of the theorem of Minkowski), and the latter generalized lemma 1. Some authors have found classes of non-singular matrices A with p ( P A , Y ) 5 2-" (detAl. Let us call a matrix A with this property a Minkowski matrix (Macbeath [49a]). If all non-singular matrices A were Minkowski-matrices, then we should have p o ( P ) = 2-", because p(P,Y ) = 2-". A first result of the type meant was given by Kowner [49a] who proved that p(PA, Y ) = 1 if det A = f1 and the parallelotope

(17)

lai,xl+

* * *

+ainx,I 5 1

(i = 1 ,

. . ., n),

which has volume 2", is an extremal convex body (sec. 12). Schneider [49a] generalized this result by proving Theorem 3. Suppose that A = ( a i j ) has determinant +1 and that the convex polyhedron M given by (18)

Iti(x)l 6 1 ( i = 1,. . ., n),

-

n

C Iti(.)[ i= 1

S tn,

where c i ( x ) = ail x 1 + * +ainxn (i = 1 , . . ., n), does not have a point u # o of Y in its interior Then A is a Minkowski matrix.

584

INHOMOGENEOUS FORMS

CH.

7

Proof. First we show that M has volume Y-'. Consider the polyhedron A M obtained by subjecting M to the transformation A x = y and denote by A , the sub-lattice of Y consisting of the points u E Y whose coordinates are all even or all odd. The polyhedron AM may be described as the set in R" obtained if the cube lyil p 1 (i = 1, . . ., n ) is truncated by the 2" hyperplanes each of which bisects orthogonally a line segment joining the point a and a vertex of that cube. From this characterization of the set AM it is clear that the bodies A M f u , u E A , , cover simply the space R". Since d(&) = 2"-' and ldet A ( = 1, it follows that V ( M ) = V ( A M ) =

- 2"-1

We now apply the remark to theorem 13.5, with H , = H , = +M and with k = 1. By the result just proved, V ( H , ) + V ( H 2 ) = 1. Furthermore, since H 1 = H2 = M , the body int H1 = int H , does not contain a point u # o of Y. Hence, by that remark, the bodies H , - H 2 + u = = M + u (u E Y ) cover the whole space. Now, since

the polyhedron M is contained in the domain n

Thus our result is that Y is a covering lattice of the domain (19). This proves that p ( P A , Y ) 5 2-". The conclusion of theorem 3 is also true if the conditions hold for some matrix DA, where D is a non-singular diagonal matrix. More generally, DA is a Minkowski matrix if A is so and D is a non-singular diagonal matrix. For we have p(PDA,Y, Dz) 5 2-" JdetDAJ if and only if p ( P A ,Y , z ) 2-" ldet Al. Likewise, AU is a Minkowski matrix if A is a Minkowski matrix and U is an integral unimodular matrix. Birch and Swinnerton-Dyer [49a] proved that B is a Minkowski matrix if PB w P A , where A lies in a specified neighbourhood of the unit matrix. (This is not a consequence of theorem 3.) Macbeath [49a] obtained a more general result, by a simpler method. His main idea is given by the following

Theorem 4. Let 0 be an orthogonal matrix and let T = ( t j j )be a triangular

8 49

585

INHOMOGENEOUS FORMS IN MORE VARIABELS

matrix with (20)

(i = 1 , . . ., n), tij = 0

t.. It = 1

if i > j .

Then OT is a Minkowski matrix. Proof. Let z be an arbitrary point of R". Choose successively the coordinates u,, u , , - ~, . . ., ul of a point u E Y such that the coordinates of the point Tu-O-'z are all S -& in absolute value. Then I0Tu-zl2 = I T u - O - ~ Z ]S~ $n. Hence, by the inequality of the arithmetic and the geometric mean, IP(OTu-z)l 6 2-". Since OT has determinant 1, this proves the theorem. By theorem 4 and the preceding remark, each non-singular matrix of the form DOTU is a Minkowski matrix. It can be shown that, in the space of all n xn-matrices, each non-singular matrix of this form is an inner point of the set of all such matrices. Therefore, each non-singular matrix DOTU (D, 0, T, U as above) has a neighbourhood consisting entirely of Minkowski matrices. Bombieri [49c] proved that, for each non-singular matrix A and each point z E R", there exists an integer k with 1 5 k 5 2*("+1), p(PA,Y,kz) 6 S 2-"ldet Al. Cassels [49a] constructs an n-ary product PA depending on a positive integer k such that p(P,, Y)/ldet A1 is arbitrarily near to 2-"; thus, if Minkowski's conjecture is true, the minimum 2-" is not isolated. Oliwa [49a] applied Cebotarev's result to systems c,(x) = u l , c,(u) = = ailu1- u i (i = 2, . . ,n), where the ui satisfy certain linear congruence relations (confer sec. 8). It is not known whether or not po(P) is positive for n > 2.

.

49.3. We now come to the study of products of complex linear forms. Let r and s be non-negative integers with r+2s = n and let the form Pr,2sbe defined as in sec. 41.1. For any non-singular linear transformation A = (aij) of R", we have (21)

Pr, 2 s , A ( x ) = Pr, ,,(Ax) = t1(x)t2(~) = t,(x). * tr(x)itr+1(x) = 2-"t1(x) tr(x) *

.

---

* {(v1(4>z

tAx> = * tr+s(x)12 =

* *

.

*

+ (cl(4)21 - { ( v s ( 4 > 2 +(cs(4)21, * *

586

INHOMOGENEOUS FORMS

CH.

7

where we have put

+ i C j ( x > > 7 t r + s + j ( x > = t r + j ( x ) ( j = 1, S) and where < , ( x ) , t , ( x ) , . . ., t n ( x ) ,in this order, are the n linear forms u j l x l + . +ujnxn ( j = 1, . . ., n). The corresponding inhomogeneous (22)

t r +j(x)

= 2-+(Vj(x)

*

*7

z ) be written as form P r , z s , A ( ~ - can

n

pr, Z s , A ( x - z z ) =

(23)

ll

j=1

(tj(xbfij17

where P I , . . .,fir are real numbers and fir+l , . . ., fin are complex numbers with f i r + s + j = ( j = 1, . . ., s). WithPr,zs,Awe associate the positive definite quadratic form n

(24)

Q ( x ) = Q r , ~ s , A ( x ) = C ltj(x)l2 = j= 1

=

j= 1

+

(
{(qj(x))2

j=1

+(lj(x))'}

=

2

j,k=l

sjkxjxk,

with some symmetric matrix (sjk).If A , A , and D denote successively the determinant of the n forms the determinant of the n forms t1, . . ., tr7q l , . . ., qs, 11,. . ., C,, and the discriminant of Q, then JA(= /All = D ' . A particular case arises if N = P,,2s, A is a norm form of an algebraic number field F of degree n. The conjugate fields consist of r real fields F(') = F, F ( 2 ) , . ., F") and s pairs of complex conjugate fields F ( ' + j ) , F ( r + s + i ) ( j = 1, . . ., s). We have

cj,

.

n

N(x) =

n

j = 1

(xl m y + ' *

*

+xnml;i)),

where (ml, . . ., con} is a basis for the algebraic integers in F and o (1) k , . . ., mp) are the conjugates of mk (k = 1, . . ., n). The discriminant of F is D' = kD, for one of the signs, D being the discriminant of the associated quadratic form. It is easy to see that po(Pr,zs)is infinite if s > 0. For take a positive number E and choose A such that tj(x) = xi

( j = 1,. . ., r),

tr+j(x) = x r + j + i c x r + s + j

( j = 1,. . .,s).

0 49

t j have determinant A

Then the forms

= +(2ie)", whereas

n

(26)

if z

587

INHOMOGENEOUS FORMS I N MORE VARIABELS

] P r , z s , A ( u - z )=~ =

($,

. . ., -5).

n

j= 1

Itj(u-z)l

2 2-"(1+e2)"

for all u E Y

Hence,

4 p r , Z s , A , y)/Idet AI 2 2-"((1 +E')/(~E))S-

Clearly, the last expression is unbounded and so ,U'(P~,,~) = 00. Geometrically speaking, the result means that for s > 0 the star body IP,,zS(x)I 5 1 has inhomogeneous determinant 0. Nevertheless, it is possible to derive an upper bound for the quantity A , Y ) . Clarke [49b] proved I@,,

,

Theorem 5. If N = P,,2s, is a norm form of an algebraic number Jield, with discrim inant D then I,

A N , Y)

(27)

s YID'ln'z,

where y is a positive constant depending on n only. Proof. Consider the form P,,zs, A . If U is an integral unimodular matrix, then p(Pr,Z s , A I I , Y ) = P ( P , , ~ Y ~ ), .~Therefore, , it is no loss of generality to suppose that the associated quadratic form Q = Q,,,,,, is reduced in the sense of Minkowski. Then the coefficients sll,szz,. . ., snn are nondecreasing. As in sec. 10, we decompose Q ( x ) as a sum n

(28)

C1Pj(xj+rj, j + l " j + l +

j=

*

*

+Yj.nXn)',

where the yj,k ( j < k) are real numbers and where Pj is a positive number 5 sjj ( j = 1, . . ., n). There are several inequalities involving the coefficients s j j . First of all, by theorem 10.3, there exists a positive constant TC only depending on n such that s1 sZ2 . . . snn 5'KD.Next, let z be an arbitrary point of R". Using the decomposition (28) of Q ( x ) we see that we can choose successively u,, u n M 1., . ., u1 in such a way that Q(u-z) 6

&(Pi+

*

. +an) S

4nsnn.

By the?inequality of the arithmetic and the geometric mean, this yields

S

~pr,Zs,A(u-~)~

(i

t n

Q(~-z)) 4

(bsnn)*"*

588

INHOMOGENEOUS FORMS

CH.

7

For a similar reason, since Pr,2s,A(e')is the norm of an algebraic integer # 0, sll= Q(e') >= n. Hence, sllsZ2. * s,, 2 n"-'s,,. Collecting the results we find that

lPr,2s, A(u- z)I 5 yD",

where y

=

(~/(4n"-'))%,.

This proves the theorem, because 1 0 ' 1 = D. By a refinement of the proof Davenport [49f] found that (29)

fl(K Y ) 5 Y

n I ( 2 n - 2s)

PI I

9

where again y is a positive constant depending on n only. Actually, Davenport constructed a non-singular diagonal matrix L with the property that the last r + s successive minima of the form Q = Q,,,,,.., i.e. the quantities AS+,(Q, Y),. . ., A,(Q, Y ) , are nearly equal. The construction of L makes use of the theory of polar reciprocal bodies and the theorem of Siegel-Davenport (theorem 24.2). An entirely different proof of (29) was given by Barnes [49b]. Below we come back to this proof. As was remarked by Davenport [49g], the relation (29) can be generalized to the class of all forms P,,2s, A . The reason for this is that the only specific property of the norm form N used in the proofs of (27) and (29) is the fact that N has homogeneous minimum Al(N, Y) = 1. Substituting p(Pr,Zs,A,Y)IA,(Pr,Zs,A, Y ) f o r ~ ( NY) , and(detA)/Al(P,,,s,A, Y ) for 1D'I3 in (29) one is led to the following

Theorem 6. Let r, s be non-negative integers with r+2s = n and s > 0, and let A be an arbitrary lattice. Then (30)

M P r , 2s 3 A ) > ~{A,(Pr, 2 s 9 A)}' a

-'6 Y ~ ( A ) ,

where 9 = 1 -s/n and where y is a positive constant depending on r and s only. Proof. (with Barnes' method). It is convenient to introduce the form ~ ( x )= 2spr,2s(x) = ~ 1 x .2. . x r{(xr+1)2+(xr+s+1)2)'..((xr+,)2+(x")2).

Let 5' denote the star body IF(x)l 1 and let z be an arbitrary point of R".For simplicity, put 1 = Al(F, A ) , p = p(F, A , z). We have to prove that

(31)

p9A1-'S $(A),

I 49

INHOMOGENEOUS FORMS I N MORE VARIABELS

589

for some constant y not depending on A and z. Clearly, it is no loss of generality to suppose that 1> 0, p > 0 (observe that the quantities 1,p are always finite). First we reduce the problem to the case that p is attained and that 1< p = 1. Take a sequence of points xk E A and a sequence of aatoof S such that IF(X-z)l + p for k 00 and that the morphisms sequence {Qk(.”-z)} is bounded. The lattices QJ all have determinant d(A) and are all admissible for the star body IF(x)l 5 1.Hence there is a subsequence {Qkr}such that {&,A} converges to some lattice A’, and that {Qk,(xki-z)}converges to a point z’, say. As in the proof of theorem 26.6, we have p(F, A’, z‘) = p, IW)I = P. --f

Furthermore, by theorem 25.6, A(F, A’) 2 lim sup I ( F , QkrA) = 1(F, A ) = 1. i-rm

Consequently, (31) holds, with some choice of y, if the corresponding relation for A’ and z’ holds true. Also, p(F, A’, z‘) = IF(z’)I. Next, we observe that (31) is not affected if A and z are subjected to any transformation aQ, where a # 0 and Q is an automorphism of S. From all these remarks we conclude that it is no loss of generality to make the following assumption z = (1, 1, . . ., 1,0, . . ., 0) ( r + s times 1) p = inf {IF(x-z)l: x E A } = IF(z)l = 1.

(

(32)

Under this assumption, (31) takes the form A’-’ 5 yd(A). Now d(A) 2 LA (S),because A is admissible for the star body IF(x)l 5 1which has critical determinant ,IA(S) (the quantity A ( S ) is finite). Therefore, it is no loss of generality to make the additional assumption that

a<

(33)

1.

We now put 6 = (4s+1)-’rZ2/”.We show that, under the conditions (32) and (33), the lattice A is admissible for the parallelotope (34)

lxtl

< 1 (i = 1 , . . ., r), I x , + ~ ~-= 1 ( j Ixr+s+jI

< JS

( j = 1,

-.

For each point x # o of A we have IF(x)l

-9

=

I,.. .,s),

s).

2 1 and IF(zj~x)l2 1,

CH. 7

INHOMOGENEOUS FORMS

590

for both signs. Thus, for each point x # o of A , r

s

IiflxiI. =1 jfl{(xr+j)’+(~r+s+j)~) =1 12

(35) r

=

S

I

r

i= I

S

(1-(XiI2)I

*

2

fl ((1-(xr+j)2)2+2(1+(xr+j)2)(xr+s+j)+

j =1

2 1.

+(~r+s+j)~>

Suppose there is a point x = (xl , . . ., x,) satisfying (34), (35) and (36). Then o 5 I-(X,)’ iI for i = 1 , . . ., P,

s

0 _I (1-(xr + j)’)’ + 2(1 +(xr+ j)’)(Xr+s+ j) + (xr+s+j) ( 1 - ( ~ , + ~ ) * ) * + 4 6 + 65~ 1+46+6’ < (1+26)’ for j = 1 , . . ., s. 2

4

Hence, by (35) and (36),

2 (1+26)-’” >= 1 - 4 ~ 6 ( i = I , . . ., s), ( 1 - ( ~ , + j ) ~ ) ~ + 4 6 +56 (1+26)-””-” ~ 2 1-4(~-1)6 ( j = 1 , . . .,s), l-(~i)’

because (1+26)“(1-2m6) 5 1 for m = 1,2, on account of (33), and so

. . ..

Now (4s+ 1)s < 1,

(1-4~6)’ < 1-8~6+4~(4~+1)6’-6’< 1-4~6-6’. Hence, (1 -(x,+~)’)’ > (1 -4sd)’ (xi)’

whence r

n

Ii = 1 xi1

s 486

( j = 1, . . ., s). So we have

for i = 1 , .

. ., r+s,

S

JJ {(xr+j)2+(xr+s+j)2}s (4sS)’r((4s+1)6)”<

j=1

((4s+1)6)*” < A.

This contradicts (35). So A is admissible for the parallelotope (34). We may conclude that d ( A ) 2 6” = ( 4 ~ + 1 ) - * ~ P ’This . completes the proof of the theorem, with y = max (d(S), (4s+ 1)*’). Theorem 6 is trivial in the case s = 0. For then not only d(A) 2 AA(S), but also d(A) 2 ,ucd’(S) >= 2*”pL,by theorem 2. Davenport [49g] illustrates by a simple example that the exponent in (30) is best possible in the case

5 49

59 1

INHOMOGENEOUS FORMS IN MORE VARIABELS

r = 0. He also shows, by choosing differently the number 6 occurring in the proof of theorem 6 , that (30) holds with 9 = 1- (2s+ 1)/(2n-2) if 0 < s < i n . A simFle application of theorem 6 is that, if F is an algebraic number field of degree n, A is an (integral) ideal in F and CI is an arbitrary number of I;, there is an integer q in F with = a (mod A ) , INm ql < yN(A)ID'J"'(Z"-2S) (Davenport [49f]). Further improvements of theorem 6 are given by Davenport and Swinnerton-Dyer [49a]. Prasad [49a] and Swinnerton-Dyer [49a] treat special cubic forms. Clarke [49b] shows that in the case r = s = 1 the best possible constant in (30) is not less than $. Finally, it should be remarked that the absolute minima ,uo(P,,,), ,uo(Po,,) are positive; as a consequence, there are only finitely many cubic fields with r = s = 1 and finitely many quartic fields with r = 0, s = 2 in which the euclidean algorithm holds. The proof of this fact is similar to that of theorem 48.4 (Cassels [48a]). 49.4. We comment briefly results for other inhomogeneous forms and

deduce a result of Rogers. Let Q be an indefinite quadratic form in n variables with discriminant D # 0. Blaney [ ~ O C ] proved that, for such a form Q, p(Q, Y ) S 2"-'1D1''". This result was improved by Rogers [49a] who derived the following

Theorem 7. For arbitrary Q with discriminant D # 0, P(Q, Y ) 5 p = +n2y,lD11'",

(37)

where yn is Hermite's constant. Proof. The form Q may be written as FA,where A is a non-singular linear transformation of R" and F ( x ) is one of the forms (38)

Qr,n-r(X) = (XI)'+

*

a

+(xp)'-(xr+~)~-

* * *

-(xn)',

1 We have D =

7.5 n-1.

(det A)'. Let A be the lattice AY. Then P(Q, Y ) = P(F,A), P = fn2Yn(d(A))2'"*

We wish to prove that p(F, A ) 5 p. In order to do this we construct at first a certain new lattice Ao. Let S be the star body IF(x)l 5 1 and let G be the group of automorphisms of S given in sec. 26.4. Consider the form Qto)given by Q'"(x) =

592

INHOMOGENEOUS FORMS

CH.

7

= Ixlz and take a sequence of automorphisms D , E G such that Q r A ) -+ 2 for r + co, where

A,(@'),

2

= SUP {Al(Q'",

QA): D E G ) .

From a certain index onwards the lattices DrA are admissible for the sphere 1x1' S 32. Therefore, the sequence of lattices Q r A is bounded, in the sense of definition 17.2, and so these lattices have cells contained in some fixed sphere K, say. Now take an arbitrary point z E R" and put r = A+z. By the foregoing,thesphereKcontainsapointzrofQ,~=QrA+Qrz,for r = 1,2, .... Since the sequence of lattices S2,A is bounded, we may conclude that there is a subsequence { Q r i } such that the sequence {QriA}converges to some lattice A , and that the sequence {O,,z") converges to some point zo. Then the sequence {DJ} converges to the grid To = A,+zO. Furthermore, d(A,) = d(A). Also, A,(Q"', A , ) = 2, because A1(Q('),A ) depends continuously on A . We now deduce the required inequality under the assumption that A , has n independent points a', . ., a" on the boundary of the sphere Q("(x) g 2. Let y be a point of roof the form 9, u1 * * Qnd, where 9,, . . . ,9, are real numbers with 19J 6 (i = 1, . . ., n). Then, by our assumption and the definition of the constant yn,

.

+

+

lyI2

5 +la'+

+a"I2 < +(lull+

= +nZA1(Q(O),A,)

+

* * .

5 ~nz~,,(d(AO))"" = p.

Hence, for some index r, there is a point y r in S2J point Qr-'yr belongs to r. Moreover,

IW&-'Y?l

=

+(a"1)2 =

with IyrI2< p. The

IF(Y31 5 IY'12 < P*

Thus f ' = A+z has a point in the domain IF(x)l < p. This being true for each point z E R", we find that p(F, A ) 6 p which is the assertion of the theorem. We now remark that, for all S2 E G, Al(Q('), L2.4,) = lim A1(Q('), lZL2,,.4)

5 1.

t-rm

Therefore, to complete the proof, it s~&ces to derive the following lemma (where we taken X = 1).

8 49

593

INHOMOGENEOUS FORMS IN MORE VARIABELS

Lemma 2. Let Knbe the unit sphere in R" and let A be a lattice withl,(Kn,A ) = 1 which has at most n - 1 independent points on the boundary of Kn. Then there exists an automorphism R E G such that Q A has no point # o in the interior or on the boundary of K,, (so that A1(Q(O), RA) > > Al(Q(0), A ) = 1). Proof. The points of A on the boundary of Kn lie in some hyperplane a1x1 . * . a,x,, = 0. Applying a suitable automorphism of S composed of an orthogonal transformation in the space of the first r coordinates and an orthogonal transformation in the space of the last n - r coordinates we obtain that the points meant lie in some hyperplane ax1+ LX'X,+~= 0, or also P(x, + x , + ~ )= P'(x, - x r + J , where a, a' and similarly p , P' are not both zero. We may suppose that IPI S Ip'I. Then

+

P'

+

# 0.

We consider the effect of an automorphism R1,,+ of the form x;+x:+~ = r ( x 1 + x r + 1 ) , xi = xi

x;-x:+l

= z-'(X1-xr+l),

(i # 1, r + 1 ) ,

where z is just greater than 1. We observe that, for x 1x1' = 3 { ( x 1 + x r + 1 ) 2 + ( x 1 - x r + 1 ) 2 ) +

C

E

R",

i# 1, r + 1

(xi)2*

The points of A outside K,, are transformed by Q1,,+ into points which lie again outside K,, provided z is sufficiently near to 1. Further, if x is any point on the boundary of Kn with /?(X,+X,+~) = P ' ( x l - x r + l ) , then Ixl-xr+lI S Ix1+xr+,I and so lQl,r+lx] 2 1x1. The equality signs hold if and only if x1- x ~ += ~X ~ + X , + ~= 0. Consequently, if z > 1 and z is sufficiently near to 1, the lattice R l , r + l A is admissible for K , , whereas the points of this lattice belonging to K , all lie in the subspace x1 = X r + l = 0. In particular, the points just considered lie in the hyperplane xr+ = 0. Repeating the argument on the variables x2 , xr+ (if r > 1) we find an automorphism R2,r+lE G such that R2,r+l!21,r+lAis admissible for K,, and that the points of this lattice belonging to K,, all lie in the subspace x1 = x2 = x r + l = 0. Continuing with suitable automorphisms Q 3 . r + l , . . ., R r , r + l , Q l , r + 2 , . . ., R1,,, we finally obtain a lattice Q1.n * *

Ql,r+2Qr,r+1

. .. Q I , r + l A

which has no point # o in Kn. This proves the lemma.

594

INHOMOGENEOUS FORMS

CH.

1

Davenport (see Rogers [49a]) remarks that the lattice A , occurring in the proof of theorem 7 may be rotated such that a’, . ., un are transformed into points of the form b’ = (bi,1, . . ., b i , i ,0, . . ., 0). Using this fact one is led to the result that p ( Q , Y ) 5 $ny,lDI””. Birch [49a] gives a natural generalization of Minkowski’s result for binary forms by showing that p ( Q , Y ) 5 (+lDl)”nif n is even and r = n/2. The equality sign holds if and only if Q ( x ) is proportionally equivalent to Q,(x) = x , ~ - , x , ~ + ~ x ~ -The , ~ ~proof . uses lemmas on asymmetric approximations and results for ternary and quaternary quadratic forms. Watson [49a] proves that non-degenerate indefinite forms Q which have integral coefficients with greatest common divisor 1 , satisfy the relation

.

zt!;’

(39)

p(Q, Y ) = O((D1’)

for 9 >

3n-4 4(n - 112 *

Davenport [49h] investigates indefinite ternary quadratic forms Q of discriminant D # 0. He shows that, for such a form, p ( Q , Y ) 5 - (271D1/100)iand that the constant 27/100is best possible. Moreover, I the value (27/100)*is isolated, in contrast with the situation for indefinite binary quadratic forms. The investigation was carried further by Barnes [49c, 49dl. Barnes [49e} can easily prove that p ( Q , Y ) = 0 if Q is a non-degenerate indefinite quadratic form in n 2 3 variables which represents arbitrarily small values of each sign. An example of such a form is where 9 has a simple continued fraction expansion with unbounded partial quotients. Barnes’ proof uses a decomposition like (44.16) and employs a result of Blaney [~OC] on asymmetric inequalities for forms in n - 1 variables. The result implies that po(Qr,n-r) = 0 for any of the forms (38) i f n 2 3 and 1 S r 5 n-1. Hlawka [49a] generalizes theorem 48.4. His result is as follows. Let f1 , f be the distance functions of bounded star bodies in R“’,R”’, respectively (nl 2 1, n, 2 1, n,+n, = n). Let S be the star body in R“ consisting of the points (x’,x ” ) with fl(x’>”1f2(x‘‘)”2 5 1. Then p o ( S ) is positive. In particular, it follows that the quantities po(P1, ,), pLo(Po,4) are positive (see sec. 49.3).

8 50

595

ASYMMETRIC INEQUALITIES

50. Asymmetric inequalities 50.1. For CT,z 2 0 let S,,,, denote the domain -C T S xlx2 S z. According to theorem 47.1, A’(S1, ,) = 4. In the general case, we have

Theorem 1. For arbitrary

CT,z

(1)

2 0,

A ’ ( & , ) 2 4(az)*.

From (1) it follows immediately that, if CT and z are positive, each grid r. of determinant d ( T ) < 4(az)* contains a point of S,,,,. Thus theorem 1 yields the following

Corollary. Let ti(x) = a i l x l + a i z x z ( i = 1,2) be two linear forms of determinant A # 0 and let CT,z be two positive numbers with CTT > ($4)’ = = &S(Q), where S(Q) is the discriminant of Q(x) = tl(x)t2(x). Then for all real a I , a, there is a lattice point u satisfying (2)

-0

6

(t,(u)-a1)(tz(u)-az)

S

7.

The corollary was proved by Davenport [49h], under the weaker condition CTZ 2 ()A)’. Later, Sawyer [50a] gave a geometric proof of theorem 1 which is very similar to Sawyer’s proof of theorem 47.1; it is based on the fact that ldet (a, b)l 2 2(az)* if a, b are any two points lying in consecutive quadrants, not inside S,,, r . The estimate (1) is not exact for all CT,z. Thus, using Delone’s algorithm of divided cells, Barnes and Swinnerton-Dyer [48a] proved that

A’(S,,,) 2 {(z+l)(z+9)}*

(3)

if z 2 3.

This is better than (1) if CT = 1 and z > 3. The authors cited derive various other estimates for A’(S,,,). For certain values of z the estimates obtained are exact. See also Blaney [50a]. Cassels [50b, ~ O C derived ] various improvements of the corollary* to theorem 1 involving values Q ( u ) of the homogeneous form Q(x) = tl(x)tz(x). Thus he found that, if Q(1,O) >= 0, there always is a solution of (2) with - CT

*

= min (0, min Q(k, l)),

We always suppose that A

= det (au)

# 0.

z = Q(l,O),

596

INHOMOGENEOUS FORMS

CH.

7

where k runs through all integers. He also gave a result which involves one value q = Q ( u ) only; in the symmetric case CJ = z, this result furnishes a slight improvement of theorem 47.7. For the case CJ = 0, see below. The method used by Cassels is a geometric one. Davenport and Heilbronn [5Oa] dealt with the case CJ = 0. First of all, they proved that there always is a lattice point u satisfying (4)

0

5 (ti(~)-~i)(Cz(u)-az) < ldl

= (4S(Q>)3.

This is even true under the additional condition that t i ( u ) - a i 5 0 for i = 1,2. Further, they showed that for each number 9 with 0 < 9 < 1 there are forms ti(.) - ai (i = 1, 2) such that one cannot satisfy (5)

05

( 4 1 ( ~ ) - ~ ( 1 ) ( < 2 ( ~ ) - - 2 )59141.

These results tell us that d’(S,, = 1. A proof of this relation, and even of an n-dimensional generalization of it, will be given in sec. 50.3. There are more special results in which there appears a homogeneous value q = Q(u), where u # 0.Let q be such a value, with 0 5 q < 1 (see the corollary to theorem 44.1). Then (5) admits a solution if we take 9 = 9,, where 9, = q if 4 5 < 1 and 9, = q + + if 0 5 q < 3 (Davenport and Heilbronn [50a]). We may even take 9 = 9, where 9, = 3 if q = 0, and 9, = qm, m the smallest integer 2 1/(2q), if q > 0 (Cassels [50b]). The last result is a special instance of a general, but more complicated, result concerning the system (2). On the other hand, Blaney [50b] proved that there is a positive constant 9, such that the system 9014

s (51(u)-a1)(52(u)-c72)

5 lA

always admits a solution and that - 11+3,/56 is the best possible value of 9,; moreover, this value is not isolated. The proofs of these results are complicated and require the consideration of many cases. 50.2. A different problem arises if one or both forms Ci(x)-ai are

required to be positive. In this case we have the result that the system

(6)

1(5i(u)-ai)(52(u)-a,)l

5 Wl

=

(S(Q>)*,

CZ(U>-~Z > 0

+

always has a solution (Cole [5Oa]). This is sharp for the form (ul $)uz. Thus, geometrically speaking, the domain Ix1x21 5 1, x2 > 0 has in-

591

ASYMMETRIC INEQUALITIES

homogeneous determinant 2. The result can be generalized to the ndimensional case (see the next subsection). We remark that ( 6 ) is solvable with $141 replaced by ldl/J5 if at least one of the forms ti(.) is not commensurable. Cassels [50a] treats the system -u 5 ( ~ ~ - $ u ~ - a ) u6, 2, where 9 is irrational. See sections 41.4and 41.7. Sawyer [50b] proves that the domain Ix1x215 1, x1+ x 2 > 0 also has inhomogeneous determinant 2. The set determined by the inequalities Ix1x216 1, x1 > 0 or x2 > 0 has inhomogeneous determinant 45. Macbeath [50a] applies theorem 35.1 to the set xlx2 2 a, x1 2 0, x2 2 0, where a > 0. He finds that, if a > 0 and IC > 2, the domain a 5 x1x2 c (K- l)'a, x1 > 0, x2 > 0 has inhomogeneous determinant 2 ~ ( I--2)(x2 C - 4)*. For integral K there is equality. By a similar method, Macbeath [50b] treats the domain T., ,:a 5 x2 - (xl)' 5 p, where a < B. He also shows that, for arbitrary p > 0, there is a finite set of lattices A , , . . ., A , such that each grid which is strictly admissible for To,,, necessarily has the form A i + (0,z2), 1 i 5 m. 50.3. Next we deal with asymmetric inequalities for forms in more

variables. We prove an elegant result of Chalk [SOa] on products of linear forms. As usual, we write P ( x ) = x l x 2 * * x,, PA@) = P(Ax), A = (aij), ti(.) = a i l x + * * * +ainxn (i = 1,. . ., n).

.

Theorem 2. The domain S+ giuen by P ( x ) <= 1, x i > 0 (i = 1,. .,n) has inhomogeneous determinant 1. Each grid of determinant 1 contains a point of S , .

Proof.* As a grid, the lattice Y is admissible for S,. So we have A'(S+) 6 1. Now consider an arbitrary S+-admissible grid r with inf{P(y): y E r, yi > 0 (i = 1,.

+.

. .,n)} = 1

and take a positive number E < The grid r contains a point z with 1 5 P ( z ) < (1 - E ) - * , zi > 0 (i = 1, . . ., n). This point z can be transformed into a point of the form (a, a, . . ., a) by a suitable automorphism

* As in the proof of theorem 49.2, one may show that there exists an S+-critical ,1). If use is made of this remark, the proof of the grid containing the point (1,1, first assertion of the theorem is particularly simple.

...

598

CH. 7

INHOMOGENEOUS FORMS

l2 of S , , We have 1 5 u < (1 - E)-"("'). We consider the grid r' = u- 'ar. Let A' be the corresponding lattice. The grid I" contains the point (1, 1,. . ., 1). Furthermore, it is admissible for a-'S+. Therefore, n

n ( l + x i ) 2 a-n > (1--&)*

(7)

i=l

for each point x E A' with 1 + x i > 0 (i = 1, . . ., n). We show that this entails that A' is admissible for the cube lxil 5 1 (i = 1, . . ., n). Suppose that A' contains a point x # o with lxil < 1 (i = 1, . . ., n). We can arrange that max [ x i [2 +. Applying (7) to the points + x we find that n

3 2 JJ (1-(xJZ) > 1-&. i=l

This contradicts the choice of E and therefore proves the assertion on A'. By Minkowski's theorem, we may then conclude that d(A') 2 1, so that d(T) = a"d(A') 2 a" 2 1. If d(T) = 1, then necessarily P ( z ) = a" = 1. From these remarks the assertions of the theorem follow.

---

Corollary. Let ti(x) = a i l x , + +ai,x, (i = 1,. . ., n ) be n linear forms of determinant A # 0 and let ai, . . ., a, be real numbers. Then there is a lattice point u satisfying

Remark. In particular, each lattice of determinant 1 contains a point of S + . Since o $ S , and Y is an S+-admissible lattice, it follows that we also have A ( S + ) = 1. Chalk also enters upon the question for which systems of forms

t i ( x ) - u ithere is not a solution of (8) with strict inequality. At any rate,

it follows from the Minkowski-Haj6s theorem (sec. 12.4) and the proof of theorem 2 that for such a system, apart from constant factors # 0 and apart from the arrangement of the forms, the set of homogeneous forms t i ( x )is equivalent to a set of forms

(9)

&(X)

=

bi'X,+

* . -

+bi,i-'Xi-l+xi

(i = 1,

..., n).

B 51

599

INEQUALITIES WITH INFINITELY MANY SOLUTIONS

Chalk proves that the coefficients bij are all rational and that, for n = 2, = a, = 0 one can obtain that b2, = 0. In the special case a1 = a2 = he finds: for n = 3, the system (9) can be so chosen that

- -

q1(x)q2(x)~3(x)= x l x 2 x 3 for n

=

Or

x1x2(~x1 +3x2+x3);

4, one can obtain that q1(x)q2(x)q3(x)q4(x) is one of the forms

x 1 x2x3 x4 >

x1 x 2 x 3 ( 3 x l

x1

+$x2

x2x3(4x1 + 3 x 2 +QX3

+x4),

+x4),

x1

x2x3(3xi + 3 x 2

x 1 x2(4x1 + 3 x 2

+x4),

+*x3

+x3)(3x1+ 3 x 2

+Sx3

+ x4)*

Another proof of theorem 2 was given by Macbeath [47a]. Cole [5Oa] proves that the domain xlxz * . X,-~IX,/ S 1, x i > 0 (i = 1 , . ., n- 1) has inhomogeneous determinant 2 and that each grid of determinant 2 contains a point of this domain. Rieger [50a] extends theorem 49.6 and Davenport’s improvement thereof to the case in which only points x with xi> 0 (i = 1, . . ., n ) are taken into account.

.

Blaney [~OC] investigates non-degenerate indefinite quadratic forms Q in n variables. He finds that for each constant y 2 0 there exists a constant y‘ depending on y and n only such that for each point z E R” one can solve The proof proceeds by induction on n and uses similar techniques as, for instance, the proof of theorem 44.4.Blaney [50d] and Barnes [50a] prove that, if Q is an indefinite ternary quadratic form of discriminant D < 0, there is a lattice point u such that 0 < Q(u-z) S (41D1)*. The equality sign is needed only if Q is proportionally equivalent to ( X , ) ~ + X ~ Xor~ (x1)2+

(x2)2-2(x3)2*

51. Inequalities with infinitely many solutions 51.1. In this section we shall make use of Kronecker’s theorem. This theorem itself provides an example of a system of inequalities in integers ul ,. . ., u, admitting infinitely many solutions. We present a short proof, but do not deal with the refinements and generalizations of Kronecker’s theorem, which have their proper place in the theory of uniform distribution. In its simplest form, the theorem of Kronecker says that, if 9 is an

600

cn. I

INHOMOGENEOUS FORMS

irrational number, the set of numbers u19-u, is everywhere dense in the set of real numbers. More generally, we have Theorem 1 (KRONECKER'S m f q = n and let

THEOREM). Let llz

and q bepositive integers with

(i = 1, . . ., q )

+ - - - +ai,x,

(1)

&(x) = a i l x l

be q homogeneous linear forms in m variables xl, . . ., x, with the property that thenforms t i ( x ) (i = 1, . . ., q), t , + j ( x ) = x j ( j = 1, . . ., rn) are independent over the ring of integers. Thenfor each E >0 and each set of q real numbers a,, . . ., aq there is a lattice point u = (ul, . . ., u,) satisfying

ltl(ul,. . ., u,)-ai-u,+i(

(2)

(i = 1 , . . ., q).


Proof. Let E~ be a positive number c c/n and let w be a positive number to be chosen below. Consider the parallelotope K given by \xjl j o ( j = 1 , .

(3)

Iti(X1,.

. ., x,)--x,+J

. .,m),

(i = 1, - . 4).

5 El

.Y

The n linear forms appearing in (3) are independent, and K is a bounded o-symmetric convex body. We show that K contains n independent points of Y, provided w is sufficiently large. (i = 1,. . ., q) Put q i ( x ) = q,(xl, . . ., x,) = t i ( x 1 , .. ., x,,,)-x,,,+~ and take an arbitrary linear form ~ ~ + ~=( klxl x ) . * +k,x, with integral coefficientsk , , . . ., k,, not all zero. If there were real numbers p l , . . . , p q such that

+-

'iS+l(X)

= Pl?l(X)+

then we should have k,+,

k,x,+

- + k,x, * *

-

* *

= -pi

= -k,,,+, <,(x,,

+P,?,(4

(XEWY

(i = 1,. . ., q) and therefore

..

a,

x,)-

* *.

-k,t,(Xl

9

. . .,x,).

This contradicts the hypothesis of the theorem. Consequently, qq+l(x) is not a linear combination of ql(x), . . ., qq(x). Now take a point a with ql(a) = *

- - = ?,(a)

= 0, ~ , + ~ ( az)

0

and consider the cylinder C(a, a), with semi-diameter 6 > 0, whose axis is the straight line containing o, a. By Minkowski's theorem, it contains a point u # o of Y for each choice of 6. But, if 6 is small enough, C(ay6)

8 51

60 1

INEQUALITIES WITH INFINITELY MANY SOLUTIONS

does not have a point u # o in the hyperplane q,+ l(x) = 0 (which does not contain the axis of the cylinder). Furthermore, if 6 is small enough, C(a, 6) is contained in the generalized prism [q,(x)] E , (i = 1, . . ., 4). We may conclude that the points of Y in this prism are not all contained in any hyperplane qq+,(x) = kl x1+ . * +k,x, = 0. Therefore, this prism contains n independent points of Y. If w is sufficiently large, these n points also lie in the domain lxjl S w ( j = 1,. . ., m) and therefore in the parallelotope K. The number w having thus been chosen, we may say that the successive minima Al(K, Y ) , . . ., A,(K, Y ) are all 5 1. It follows then from Jarnik’s relation (theorem 16.5) that p ( K , Y ) 5 +n < n. Thus Y is a covering lattice of nK. In particular, it follows that for each point z = (0, . . ., 0, a,, . . ., a,) there is a point M E Y with u-ZE: nK. This point u satisfies (2), because nc, < E . This proves the theorem.

Remark 1. The assertion of theorem 1 also holds if the set of forms t i ( x ) ,xi is not required to be independent over the ring of integers and the set of numbers ai satisfies the condition that kl a, * * +kqaq is integral for each set of integers kl , . . ., k, such that the form k , ( , ( x ) + + . . . +k,(,(x) has integral coefficients. This is easily deduced from theorem 1.

+

Remark 2. From the last part of the proof of theorem 1, in particular the fact that Y is a covering lattice of the bounded domain nK, it follows that there is a lattice point u satisfying (2) and lying in any given half-space PI x1 + . . . + Pnx, + a L 0. (Translate z , u over a suitable vector in Y ) . Theorem 1 can be put in a more geometric form. We consider the case m = 1. First we deduce the following

-

Theorem 2. Let q i ( x ) = b , , x , + +b,,x, ( i = 1 , . . ., n ) be n independent linearforms in n variables x l , . . ., x,. Suppose that the straight line L given by q l ( x ) = = V,,-~(X =)0 is not contained in any hyperplane k , x 1 + * * +k,x, = 0 ( k , , . . ., k, integers not all zero). Then for each set of real numbers a t , . . ., a, and each positive number E , the semi-infinite cylinder

-

(4)

---

Iqi(x)-uil < E

contains a point u E Y.

( i = 1,. . . , n - I ) ,

qn(x)-a,

2O

602

INHOMOGENEOUS FORMS

CH.

7

Proof. Permuting coordinates and replacing ql,. . ., q n d 1 by suitable linear combinations we can reduce the theorem to the case that qi(x) has the form sixl -xl + i , for i = 1, . . ., n- 1. In this case, the condition imposed on L simply says that 9, , . . ., 9,-1 , 1 are rationally independent. This is the condition of theorem 1 , in the case m = 1. The theorem now follows from theorem 1 and remark 2. Now apply the transformation y = Bx, where B = (bij).Put A = BY, C = B* = ( B t ) - ' . The straight line L is transformed into the line y 1 = . = Y , , - ~ = 0, and the hyperplane k l x l * * +knxn = 0 into the plane (Cv)- y = 0, where v = (kl, . . .,k,,). The last plane is an

-

+ -

(n - 1)-dimensional linear subspace of R" generated by n- 1 independent points of A = BY. Each such subspace arises in this way. So we have

Corollary. Let A be a lattice with the property that the x,,-axis is not contained in any ( n- 1)-dimensional linear subspace of R" generated by points of A. Then each semi-injinite cylinder with a one-dimensional axis parallel to the (positive) xn-axiscontains apoint of A. 51.2. Another example of an inequality for an inhomogeneous form

fulfilled for infinitely many lattice points u is provided by the corollary to theorem 47.3. The form in question is the product of two inhomogeneous linear forms (in two variables) satisfying certain conditions. There are similar results for products of n linear forms in n variables. A first result in this direction which deals with positive values of linear forms was obtained by Chalk [51a]. This result can be stated geometrically as follows.

-

Theorem3. Let S+ denote the domain x1x2 * * x, 5 1, xi >0 (i= 1 , ..., n). Let a be a positive number and let f be a grid of determinant 1 whose lattice does not have a point # o on the x,-axis. Then r contains a point X E S , with x l x 2 x,,-~< a. Consequently,f o r each a > 0, there are injinitely manypoints x E r n S , with x l x z * x , - ~< a.

-

-

Proof. For a > 0, denote by T, the semi-infinite hyperbolic cylinder X ~ X ~ - - ~ X , , - ~ x < i ~> ,O

( i = 1,..., n).

Let r be a grid of determinant 1 satisfying the hypothesis of the theorem

603

INEQUALITIES WITH INFINITELY MANY SOLURIONS

and let A be the corresponding lattice. First we prove that, for arbitrary a > 0, the set r n Ta is not empty. Take a point z E I' with zi > 0 ( i = 1, . . ., n) and denote by M the smallest linear subspace of R" which contains the xn-axis and is generated by points of A . This subspace has a dimension r 2 2. We consider the linear variety M' = M + z . The intersection r n M' is an (r-dimensional) grid in M'. By the construction of Mya straight line in M' which passes through a point of r n M' and is parallel to the x,,-axis, is not contained in a hyperplane in M' generated by points of r n M'. Further, T, n M' contains a semi-infinite cylinder with a one-dimensional axis parallel to the xn-axis.Therefore, by the last corollary applied to the r-dimensional space M ' , the cylinder T, n M' contains a point x E r n M'. This is a point of r n T,. Next we prove that the quantity

-

p(a) = inf (xlxz * x,: x E r n T,) (a > 0 ) (6) is bounded by a positive constant only depending on n. Let ct be fixed and suppose that p ( a ) > 0. Take a positive number E < 4 and a point z E r n Tawith z l z z * . * z, < p ( a ) / ( l - E ) . By the linear transformation

xi

(z fixed; i = 1,.

= ziyi

. ., n )

'

the grid r is transformed into a grid r' of determinant ( z l z 2 . zn)which contains the point b = (1, 1 , . . ., 1) and which satisfies the condition YlYz--=Y,2

z1z2

* *

- z,

if y

E

f' n T,. , where a'

a

=

z1 z2

zn-1

The last condition implies that (7)

y l y z - * - y , >1--E

ifyEr'nT,,

because z1z2 . zn < p(ct)/(l - E ) and zlz2* * * znz E T,. We deduce from (7) that the lattice A' of

5

r'

on account of is admissible for

cc,

the cube lxil 5 n-" ( i = 1,. . ., n). Suppose A' contains a point x # o with lxil < n-" (i = 1, . . ., n). We maysupposethatx,+ +x.-~ 2 Oandthatx, = ma x (x i,. . .,x,,-~). Then, if k is a positive integer with 1-kxi > 0 (i = 1, . . ., n), n- 1

(8)

i= 1

k n-1

n- 1

604

INHOMOGENEOUS FORMS

CH.

7

and so b - kx E Tl (note that the two inequalities in (8) cannot both hold with the equality sign). We also have b - kx E r’. Hence, by (7) and (8), the point x has the property that (9 )

l-kx,>

l-E

ifl-kx,>O

for i = 1, ..., n.

We now consider three possibilities. a) x i = 0. Then x1 = * * = x , - = O . Thus x is a point of the x,-axis. But this is excluded by our hypotheses. So a) does not occur. b) x 1 > 0 and x, 2 x l . Then I - x , > 1- E , because of (9) and the fact that lxil < 1 for i = 1, . . ., n. Thus x, < E . We also have x, 2 x1 > 0. So we can choose a positive integer k such that 4 5 kx, < 1. Then l - k x i > 0 for i = 1 , . . .,n. Hence, 1-kx, > l - E . But, on the other hand, 1-kx, 5 +. It follows that b) does not occur. c) x 1 > 0 and x, < x i . In this case we choose k such that

1

1 5 k < -.

X1

X1

0 < - -1

Then we have

hence

n n

(1 - k x i )

i=l

< (n- 1)”-2(xl+Ix,l) < 2(n- l r - 2 n - ” 5 3.

But this contradicts property (7), because b - kx E f‘ n T , . Hence c) does not occur. This completes the proof of the italicized assertion. Since zlz2 * z, 2 p ( u ) and d(T’) = (zlz2 * z,)-’, we may conclude that

- -

--

p(c1) I; I / ~ ( T ’ )5 nn2. (10) The proof can now be completed by a similar argument as the proof of theorem 50.2. It is clear that p ( u ) is a non-increasing function of CI. Put p = limz+op(u). Then p is finite. We may suppose that p is positive. Now choose a positive number q < and take positive numbers u, E with

(1 1)

(l-&)p(2n-b) > (l-q)*p.

§ 51

INEQUALITIES WITH INFINITELY MANY SOLUTIONS

605

-

As before, let z be a point of r n T, with zlzz * z, < p(u)/(l - E ) and, with this choice of z, let r' and A' have the same meaning as above. We prove that then A' is admissible for the cube lxil 5 1 (i = 1,. . ., n). Suppose that A' contains a point x' # o with lxil < 1 ( i = 1 , . . ., n). We may assume that max 2 -4. Let x be the point (zlx; , . . ., znxA). Thenz+xisapointofT. Furthermore,sinceO < l+xi < 2 ( i = l,...,n), n- 1

zi+xi > o (i = I, . . ., n),

i=l

- -

(zi+xi) < 2 n - 1 ~ 1 ~ 2z,,-~

s 2"-'a.

Thus z + x E r n T2,-,,. Hence, n

n

i= 1

so that

(zi+xi) 2 p(2n-1a),

The same estimate holds for the point 50.2, we now have

-XI.

As in the proof of theorem

n n

3 2 i = 1(1 -(4*) > (1 -s), which contradicts the choice of q . This contradiction proves that A' is admissible for the cube lxil 6 1 ( i = 1, . . ., n ) . It follows that

-

1 = d ( r ) = zlzz

-

*

znd(r') 2 zlz2 *

- 2,.

Thus zlzz . * zn 1. Also, z1z , * * * zn- 5 u. Since a may be chosen arbitrarily small, this proves the theorem. Arithmetically speaking, we have the following

Corollary. Let tl(x), .. ., tn(x)be n homogeneous linearforms in n variables of determinant A # 0 andlet a1 , . . ., a, be arbitrary realnumbers. Suppose that the equations t l ( u ) = * * = 5.- l(u) = 0 only admit the trivial SOlution U = 0. Then,for arbitrary E > 0, there is a lattice point U satisfying

-

{&(u)-ai > 0 I n

(i = 1,.

.., n)

n- 1

Consequently, for each E > 0, there are infinitely many solutions of (12).

606

INHOMOGENEOUS FORMS

CH.

7

We mention some further results. Cole [5Oa] and Chalk [51b] deduce analogues of theorem 3 for the domain x1x2 * x,- llx,,l 6 1, xi > 0 (i = 1,. . ., n - 1) and the domain lxlxz . * * x,,l 5 1, respectively. One needs here the additional condition that r has no points on the x,-axis. Rogers [51a] derives the following more general result. Theorem4. Let S be an open set inR" containedin some domain Ix1x2* * x,J5 5 jwhich is invariant under all transformations xi = mixi (i = 1, . . ., n; wl,. . ., w,,positive numbers with w1w2* w,,= 1 ) . Let r be a grid with determinant d ( r ) < A'(S) having the property that r n S has no point on the x,-axis and that the lattice of r has no point # o on the x,axis. Then, for arbitrary E > 0, r n S contains infinitely many points x with I x l x z . * * xnW11< E .

-

Further, Rogers proves Theorem 5. Let S satisfy the same conditions as in theorem 4.Let r be a positive integer < n andlet r be a grid with determinant d ( T ) < min (A(S), A'(S)). Denote by M the smallest linear subspace of R" which contains the subspace x1 = * = x, = 0 and is generated by points of the lattice of r. Suppose that either M = R" or the following conditions are fulJile& M # R", r contains a point of M , and S contains a point x with x1 = * * * = = x, = 0. Then, for arbitrary E > 0, r n S contains infinitely many points x with (xl)' + * + ( x , ) ~< 2.

- -

-

If o E S, then the conditions imposed on M are necessary. The proof of theorem 5 makes use of the theorem of Siegel-Davenport (theorem 24.2); it proceeds by induction on n and works with sequences of grids Q,r,where the Q, are automorphisms of S. The condition that r contains a point of the subspace M defined above can be put in a different form, as follows. Let r = AY+z. Put B = A* and use the letter y (instead of x ) to denote the generic point of R". Let

ti(.)

= ailxl

+ -

+ai,xn

( i = 1, . . ., n),

(aij) = A.

The (n-1)-dimensional linear subspaces of R" generated by points of A = AY are given by the hyperplanes (Bu) y = 0, tr a point # o of Y, and M is the intersection of the hyperplanes (Bv) y = 0 containing the

-

-

607

INEQUALITIES WITH INFINITELY MANY SOLUTIONS

subspace y 1 = = yr = 0. Thus M is the intersection of those hyperplanes xlyl . * xryr = 0 for which I C y~ 1 * * * xryr has the form (Bu) y , i.e. those for which there exists a lattice point u # o in Y such that

+

-

x1tl(x)+

(13) In

+

particular,

*

+

+

*

+K,~,(X)

-

= (Bv) ( A X ) = 2) *

+

X.

R" if and only if no linear combination is a form with integral coefficients not all zero. In the general case, let T be the set of points 2) # o of Y such that v * x has the form x1 (,(x)+ * +K,&(x). Let L(T)be the linear subspace of R" spanned by T. Then we can say the following. If M contains a point A u + z E r, then (Bu) . z is an integer for all u E T, because (Bv) * (Au) = = u * u is always integral. Conversely, if (Bv) z is an integer for all v E T, then there exists a point u E Y with v * u = - (Bu) * z (v E T), because dim L(T) 5 r < n. Then (Bv) * ( A u + z ) = 0 for all v E T, so that A U + Z E T . Thus we see that M contains apoint of r if and only ifxlzl * +x,zr is an integer for all sets of real number& xl, . . ., xr such that x1 t1( x ) * * * xr <,(x) is aform with integral coeficients. Using this result and applying theorem 5 to the domain S = int S+ we obtain K~ (,(x)+

* *

M

=

- +x,(,(x)

--

-

+

+

-

+

Corollary. Let tl(x), . . ., tn(x) be n homogeneous linear forms of determinant A # 0 and let u l , , . ., u, be n real numbers. Further, let r be a positive integer < n. Suppose that lAl < 1 and that no linear combination K~ t1(x)+ * * + x r t r ( x ) is a form with integral coefficients not all zero. Then, for each E > 0, there are infinitely many points u, satisfying

-

(14)

I

t,(u)-u,

> 0 (i

= 1,.

. .) n),

n(&(u)-cci) < 1,

i=1

This corollary may be seen as a refinement of Kronecker's theorem. There is a similar application of theorem 5 to the domain lxlxz * * * xnl< 1. 51.3. There is a certain connection between the approximation properties of a set of homogeneous linear forms and those of the corresponding sets of inhomogeneous linear forms. As we shall see, this connection arises from inequalities between the successive minima and the inhomogeneous

608

cn. 7

INHOMOGENBOUS FORMS

minimum of an arbitrary bounded o-symmetric convex body in R" with respect to the lattice Y. Roughly speaking, if the homogeneous forms only admit bad approximations to zero, then all corresponding sets of inhomogeneous forms admit good approximations to zero, and conversely*. A result in this direction of a certain generality was obtained by Hintin [51a] who proved

Theorem 6. Let 9,, , . ., 9, by m 2 1 real numbers. Suppose that there exists a constant y > 0 with the fol~owingproperty: for no real number z > 1 there are integers u l , . . ., satisfying (15)

IUm+13i-uil

6 l/z

( i = 1,. . ., m), 0 < u,+1

5 yz".

Then there exists a constant y' > 0 such that, for arbitrary a > 1 and for each set of real numbers a l , . . ., urn,there are integers v l , . . ., with

(16)

Ju,+,~~-u~6 - c l/a ~ ~ ~ (i

=

1,. . ., m), 0 < v,+, S fam.

Conversely, if such a constant y' exists, then there is a constant y > 0 having the aforementioned property.

Simple proofs of this theorem were given by Mahler [13a] and Mordell [sla]. The latter author operated with a so-called additional variable (confer sec. 36.4). Here we deduce theorem 6 by applying the theory of successive minima and by using the inequalities of Jarnik (theorem 16.5).

Proof of theorem 6. Put n = m + 1. Suppose that the system (15) does not admit a solution for any z > 1 and some constant y > 0. Let z > 1 be Axed and consider the parallelotope K given by 1xi-9x,1

6 l/z (i

. ., n-I),

= I,.

1x.l 5 yz"-'.

Denote by A1, . . .,L, the successive minima of K with respect to the lattice Y . The paraIIelotope I( has volume y2". It does not contain a lattice point u # o with u,, = 0, because z > 1. Hence, by our assumptions, it does not contain a lattice point u # o whatsoever. So we have A1 > 1. Now, by Minkowski's second theorem (theorem 9. I), n:-%J2"

=< A,&

- - - 4 y 2 " 5 2".

* In sec. 47.7 we discussed more special results in the case m = 1 . See further Koksma [DA].

4 51

609

INEQUALITIES WITH INFINITELY MANY SOLUTIONS

Hence A,, < l/y and so, by theorem 16.5, p ( K , Y ) 2 +n& < n/(2y). Consequently, for arbitrary real numbers a1 , . . ., a,- , there is a lattice paint v with

(u,9,-vi-ail < n/(2yz) (i = I , . . ., n-I),

~ u , , - + n z ~ - '< ~&n~"-~

These inequalities have the form (16), with a = 2yz/n and y' = - nn(2y)-("-'). The first assertion of the theorem has now been proved. Conversely, suppose there exists a constant y' > 0 such that the inequalities (16) have a solution for each set of real numbers 0 , a1 , . . .,a,with a > 1. Then it is also true that, for arbitrary a > 1 and each set of real numbers aI,. . ., a,, one can solve

1 ( ~ , - a , , ) 9 ~ - u ~ - ~ ~ ~l/a l (i = 1 , .

. ., n - I ) ,

0 < u,-an 5 fa".

For we may suppose that 0 5 a, < 1; in this case, the inequality 0 < u, 5 5 y'a" entails 0 4 v,- a,, 5 fam, whereas (0,- a,)$,-v,- t l i may be written as ~ , , 9 ~ - - v ~ - c r : , a; = cli+annSi ( i = 1 , . . ., n-1). Denoting by K the parallelotope lXi-9,Xn)

5 l/o (1

= 1, .

. ., n-1),

Ix,l

s y'orn/2

we may cmclude that p(K,Y ) 5 1. This yields a,, S 2. By theorem 9.2, we also have

ala:-l

Hence A1 follows.

2 a,a,

*

- - a, 5 f/(2n!).

2 y'/(2"n!). From this the second assertion of the theorem

Remark. From the proof of the theorem it is clear that the first assertion also holds with (16) replaced by (16')

lu,,,+18i-u,-ail S l/o (i = 1 , . . ., m), #< I

5 /l+y'o'",

where p is an arbitrary real number. Theorem 6 may be generalized for general systems of linear forms. A somewhat similar result was given by HinEin [51b]; the proof of this result makes use of the reciprocal system of linear forms. Jarnik [5la] proved a theorem on pairs of reciprocal systems which involves an arbitrary approximation function. Before stating and proving this theorem we introduce some notations.

610

Let 0 (17)

INHOMOGENEOUS FORMS

=

( S i j ) be a realp x q-matrix. For

~(0 a, , 9 ) = min max 19i,ul+ i = l , ...,p

CH.

7

5 > 0 and a E R P put *

-

*

+9iqUq-uq+i-a,l,

where the minimum is taken over all lattice points u # o in R" with luil 5 5 (i = 1, . . ., q). It is clear that this minimum exists. For 5 > 0 and b E Rq let ~(0'~ b, 5) be defined similarly. Next, let = q(9) be a positive continuous function defined on some interval 5 2 to> 0 with the propertythat, for some E > 0, ~ ( t ) < - ~an i sincreasing function of Denote by t,b the inverse of the function q. With these notatiobs we have

r.

Theorem 7. Suppose that lim infc,,

q ( t ) x ( O ,0,5) > 1. Then

lim SUP {*(5) SUP x(@, beRq

<+a

bY 5))

does not exceed a positive constant CI only depending on n and E . Proof. The quantity ~ ( 0 ~5)0 is, a non-increasing function of 5. Therefore, the hypothesis of the theorem implies that x(O,o, 5) is positive for all 9. The functionq(5) may be extended to a continuous strictly increasing function (again denoted by q) on the interval [0, a)such that cp(0) = 0. For z > 0 let K,, L, denote the parallelotopes given by (18)

pi,x,+

(19)

( i = 1,. . ., q),

6 2p

lXil

+9i,x,+xq+il

* *

IYj-(9,jYq+,+

Ivq+jl S z4

*

*

*

5

+9pjYn)I

( j = 1,

2-4

(i = 1, . . ., p )

5 z-'

( j = 1, *

. . ., P),

- ., 4),

respectively. Further, let KT be the polar reciprocal body of K,. Then K: c L, and so A,,(L,, Y ) 5 A,(K:, Y ).Hence, by theorems 14.5 and 16.5,

(20)

P(&

5

Y ) S $n&(&

7

Y ) 6 &n&(Kr, Y ) 5 $n !n/Al(Kt; Y).

It is convenient to put A(z)

= A,(& Y). The argument is now as follows. By Minkowski's theorem, A(z) 5 1 for all z 0. Furthermore, by the definition of the quantity A(z), there is for each z > 0 a lattice point u = u(') # o with

(21)

lUil

5 2"(2)

)91,u,+

*

-

*

(i = 1 , . . ., q), +9jpuq+uq+il5

=-

PA(2)

( i = 1,

. . .,p).

8 51

61 1

INEQUALITIEX WITH INFINITELY MANY SOLUTIONS

--

Hence we cannot have Si,u,+ + S i q y + u q + I = 0 for i = 1 , . . . , p , because x ( O , o , zpA(z)) is positive. Hence, if t p A ( z ) were bounded for arbitrarily large values of z, then the quantity Z - ~ A ( T )would be greater than some positive constant for these values of 7. This is impossible, because r-qA(z) tends to zero for z + co. Therefore, zpA(z) + co as z + co.

Now x(O,o, z p l ( z ) ) S T - ~ A ( T )for all z, and cp(C)x(O, 0 , t) > 1 for sufficiently large 5, by the hypothesis of the theorem. Hence there exists a positive constant zo such that for z 2 zo..

cp(+'A(z))z-q~(z) > 1

Writing cp,(t) = t v ( t ) and denoting by rl/, the inverse of the function we have

'pl

'pl(zpA(z)) > zn for z

or also A(z) > z-"$,(z")

z0. Applying (20) we find that p ( L T ,Y ) 5 $n!nzP/$,(zn)

for z 2 zo.

Consequently, for each number z 2 zo and each point b E Rq, there is a lattice point u satisfying

0 < uj 6 n!nz"/$,(z") Putting q = r n and

t

( j = 1,

. . .)p )

= I),(?) we have

Thus our result is that, for sufficiently large there is a lattice point u such that

CJ

and each point b E Rq,

0 < uj 5 u ( j = l , . : ., p), I81jul+

.

* * +Qpjup-up+j-bjl

where we have put

6 CO(CJ)

( j = I,.

. ., 4),

612

cn. I

INHOMOGENEOUS FORMS

Since a"'/$(a)

is an increasing function of

hence W(B)

if

B

c1

= (n!n)l+l/e.

B

for

B

2

cp({,),

we have

< (?l!n)'"'I/J/(o)

is sufficiently large. This yields the assertion of the theorem, with

There is a similar result with lim inf and lim sup interchanged (Jarnlk [51b]). Jarnik [51a, 51b] also derives metrical theorems. HinEin [Slc] considers a non-decreasing positive continuous function q ( z ) and one fixed point b E Rq; he proves that there are positive constants y, y' such that

for all z > 0 ZX(@, b, ycp(z)) < y' (23) if and only if there exists a positive constant y" satisfying if z, = 0 (24)

5

II j = 1 b,u,ll

*

J/l(Pu/zu)

5 YffP"

if

7"

'0,

where jIgll is the distance of { from the nearest integer, of z(p(z) and p, = max lu,l,

.

j = 1,. ..q

zu = max 19,,u1+ i = l , ...,p

The case that q(7) + 00 for z -+ Kronecker's theorem.

GO

is the inverse

- - +9,,uq+u,+d.

yields a quantitative version of

SUPPLEMENT TO CHAPTER 7 In sections xv-xvii inhomogeneous problems are considered. The final section xviii gives an account of geometry of numbers in spaces different from R". Section xv presents newer results on covering with balls. The geometric background (in the space R f n @ + l ) )of the problem of locally thinnest lattice coverings with balls and multiple coverings (mainly in the plane) are discussed. Section xvi exhibits the great advances on the conjecture of non-homogeneous linear forms, and inhomogeneous problems for indefinite quadratic forms are treated in section xvii. In section xviii a short survey is given of analogues of classical results of the geometry of numbers in spaces such as vector spaces over algebraic number fields and non-Archimedean fields or in topological groups.

xv. Covering with balls We consider coverings of R" by equal balls. The main problems consist of determining and of estimating the minimum density of an arbitrary covering or a lattice covering and of describing the set of centres of the balls in such coverings. It might seem that these problems are just inhomogeneous analogues of corresponding problems for packings. But there are considerable differences. Surprisingly, the theory of covering with balls is of a much more recent date than that of packing of balls. Estimates for the density of thinnest coverings with balls and criteria for the (local) extremality of particular lattice coverings have a different form. The existing knowledge has been enlarged considerably during the last decades. In this section we shall deal with properties of (locally) extremal lattice coverings by balls, present exact results for n 5 5 and estimates for arbitrary n, and discuss covering properties of particular lattices or types of lattices. We shall also discuss multiple coverings and relations between relevant inhomogeneous and homogeneous minima. xv.1. The (Euclidean) unit ball in R" is denoted by K , . The definitions of a lattice covering or an arbitrary covering of R" with balls of unit

CH. 7

INHOMOGENEOUS FORMS

614

radius (or, more generally, with equal balls), the density of such a covering, and of a covering lattice of K , are taken from sect. ix. The minimum density of a lattice covering of R“ with equal balls is denoted by 9(K,) or shortly by 8,; that for an arbitrary covering with equal balls is 8: = 9*(K,). Clearly,

0, 1 0: 1 1

for n

=

2,3, ....

Let A be a lattice in R”. If r~ > 0 is such that {OK,+x: x E A } is a covering of R“, the density of the covering is equal to rYV(K,)/d(A); see sect. viii.1. Since the minimum of these d s , the so-called covering radius a(A) of A, equals the inhomogeneous minimum p(K,,A) of A with respect to K,, we obtain

where 9 denotes the space of lattices in R“. If we normalize the inhomogeneous minimum by setting

the following expression for On resblts: (2)

8,

=

V(K,,)min{M(A)”:A ~ 9 } .

We express this in terms of quadratic forms. For any lattice A consider a basis A and the associated metric form Q, where Q(x) = IAx1’. The discriminant D(Q) and the inhomogeneous minimum p(Q) of Q clearly satisfy

WQ) = d(A)’,

dQ) = AK,,

A)’,

and for the normalized inhomogeneous minimum

we have N(Q) = M ( A ) 2 . Using this relation one can show that

(2’)

8,

=

V(K,)min{N(Q):”: Q ~ 9 } ,

§ X"

COVERING WITH BALLS

615

where 9 is the cone of positive quadratic forms in R N , N = in(n+ 1); see sect. v. The quantities M ( A ) and N ( Q ) depend continuously on A and Q, respectively and the minima in (2) and (2') are attained. These remarks show that for the determination of 0, we can use lattices or quadratic forms whatever is more convenient. xv.2. As a first step towards the determination of 0, we indicate a method to find all local minima of M ( A ) in 2 or, equivalently, all local minima of N ( Q ) in 9.Barnes and Dickson [xva] and Delone et al. [xva] showed that in principle this can be achieved by use of a tiling of 9 introduced by Voronoi [12a, 39al. (Incidentally this tiling was mentioned in sect. v.5.) We give an outline and suggest that the reader takes first a look at sect. vi.1 and vi.2. Let Q be a positive quadratic form. Then there is a lattice A having a basis A such that Q ( x ) = JAxI2.A and A are unique up to rotation. We say that A and A correspond to Q. For an integral unimodular transform U of R" the mapping which assigns to each quadratic form Q €9the quadratic form R , where R ( x ) = Q ( U ( x ) ) ,is called an equivalence of 9. In analogy to sect. vi.2 we say that two positive quadratic forms Q , R €9 are of the same L-type if there are lattices A and M corresponding to Q and R, respectively which belong to the same L-type (of lattices). Clearly the L-types (of quadratic forms) form a partition of 9,but it is not this partition which we have in mind. An L-type of quadratic forms is primitive if the corresponding L-type of lattices is primitive. Any primitive L-type of quadratic forms is a union of open polyhedral cones in 9 with apex at the origin. (Two quadratic forms Q , R E 9 belong to the same cone if there exist a lattice A and a basis A of A corresponding to Q and a lattice M and a basis B of M corresponding to R such that the following holds: The L-tilings of A and M are both simplicia1 and there is a linear transformation of R" which maps the L-tiling of A onto the L-tiling of M and which at the same time maps A onto B.) The closures in 9 of these convex polyhedral cones form the desired tiling of 9.It has the following properties : (i) The tiling is facet-to-facet. (ii) The equivalences of 9 map any cone of the tiling precisely onto the cones which are derived from the same primitive L-type.

616

INHOMOGENEOUS FORMS

CH

I

(iii) There is a set of finitely many inequivalent cones in the tiling such that any quadratic form in 9 is equivalent to a quadratic form in one cone of the set. The number of these cones equals the number of primitive L-types of lattices (see sect. vi.2.). Barnes and Dickson [xva] and later in a more geometric way Delone et al. [xva] (see also Delone and RySkov [va]) proved that in the interior of any cone of the tiling of 9'just described there is at most one quadratic form which provides a local minimum of N . In the search for these local minima we may restrict ourselves to a set of finitely many cones, see (iii). A form in the interior of a cone of the tiling, which provides a local minimum of N has the property that each equivalence which maps the cone onto itself is an automorphism of the form (Barnes and Dickson [xva]).

xv.3. The simplest example of a form which provides a local minimum for N for general n is the form cpo given by po(x) = n

1( x i ) 2- 2 1 x i x j I

i c j

(Bleicher [xva], GameEkii [xva, b], Barnes and Dickson [xva]). This form belongs to the interior of the cone of the tiling which consists of the forms

where xo = 0 and pij L 0 for 0 S i S j < n. This cone is usually called Voronoi's principal domain. The form q0 is a multiple of the adjoint of Voronoi's first perfect form F o or of Korkin and Zolotarev's form U , (sect. v.5 and 39.5) and admits a large group of integral automorphisms, in accordance with the general theory. The density of the corresponding covering is

For n = 2,3,4 there are 1, 1, 3 inequivalent cones in the tiling of 9, respectively. For n = 4 each of the three cones provides precisely one local minimum of N . These forms and the associated covering densities

§ X"

COVERING WITH BALLS

617

were determined by Delone and RySkov [xva], Baranovskii [xvb, c], Dickson [xvb], and Barnes and Dickson [xva]. It turns out that for n = 4 the form cpo provides the minimum of N . Consequently, (4) gives the value of 8,for n 5 4. RySkov and Baranovskii [xva] showed that 'po provides the absolute minimum of N also for n = 5. Thus we have the following list (see also sect. 22.4 and 49.1):

13,

=

2n/(3$)

= 1.209199...,

=

5nJJ/24

= 1.463503.. .,

e4 = 2nl/(5Js) 6,

=

=

1.765528.. .,

5. 72n2J105/(24. 36) = 2.124285.. . ,

General upper estimates for 8, are discussed in the main text (sect. 21 and 49). New forms which provide local minima of N for arbitrary n were given by Barnes and Trenerry [xva] and RySkov [xva] (three new forms for n 2 6). These forms however lead to larger covering densities than cpo. On the other hand y o does not provide the absolute minimum of N if n is sufficiently large as follows from (4) and the estimates in sect. 21, in particular (21.24). So there are more economical lattice coverings than those connected with cpo. An effective construction based on a method of Davenport [49a,b] was given by RySkov [xvb]. Bambah and Sloane [xva] showed that already for n 1 2 4 there are more economical lattice coverings than those associated with p0. xv.4. In this subsection we collect various results connected with coverings of balls. Conway, Parker and Sloane [xia] and Bambah and Sloane [xva] determined the covering radius of several higher dimensional lattices, including the Leech lattice. It has been conjectured that the covering radius o ( A ) of a lattice A which contains n independent vectors # o of minimum length satisfies a(A)

s )fid(A)'/,

(cf. sect. xvi.1). This conjecture was proved by Cleaver [xvia,b] for = 5,6. Cleaver investigated the

n S 4 and by Woods [xvia, b, c] for n

618

INHOMOGENEOUS FORMS

CH

7

possible configuration of the minimum vectors and Woods applied the reduction method of Korkin and Zolotarev (sect. v). Some results deal with the relations between the inhomogeneous minimum p = p ( K , , A ) ( = o ( A ) )and the successive minima Ai = Ai(K,, A ) for 1 5 i 5 n of a lattice A with respect to the Euclidean unit ball K,. In sect. 13 several results of this type are derived when instead of K , we consider an arbitrary o-symmetric convex body. By applying Mordell's method of the additional variable (cf. sect. 36.4) it' was proved by Cassels [GN], p. 321 for n = 3 and by Bantegnie [xva] for general n that

<

&

= 2(2Ll(K,+ 1))1'"

. d(A)""d,

(A, ' . . A")l',

If A1 = ... = A, this is not best possible, at least not for the case n 2 6 ; see the result of Cleaver and Woods. RySkov [XVC] considered

and proved that

An entirely different and interesting estimate was given by Lenstra [xv] who compared p = p ( K , , A ) and A* = A 1 ( K , , A * ) , where A is any lattice and A* its polar lattice. He showed that

Here yl, y2, . . . , y n are the Hermite's constants in dimensions up to n. Note that we do not have an estimate of this type for PA. Densities of partial coverings were investigated by Erdos, Few and Rogers [xva], and Rogers [xva] and Melnyk [xva] studied coverings of spheres with spherical caps.

xv.5. Multiple coverings of R" by unit balls, lattice coverings as well as non-lattice coverings, were studied by Blundon [xva, b], Danzer

9: xvi

619

PRODUCT OF INHOMOGENEOUS LINEAR FORMS

[xva], G. Fejes Toth and Florian [xi.], and G. Fejes Toth [xia] in the case n = 2 ; by Few [xva], Yang [xva] and Dumir [xva] for n = 3 ; by Florian [xva] and G. Fejes Toth [xva] for arbitrary n (cf. the surveys by G. Fejes Toth [viiib] and by Erdos, Gruber and Hammer [SUP]). The results give explicit constructions of covering lattices, upper and lower estimates of densities, and an asymptotic relation. Let O n , k = 9k(K,) denote the minimum density of a k-fold lattice covering of R" by the unit ball K , and, correspondingly let O:., = 9:(K,). Then 02,2 =

202, d2* < k0,

for k 1 3

Blundon [xva], Danzer [xva],

G. Fejes Toth and Florian [xia]

-

0 3 , k< k03 for k 2 2

0,. k/k

1 as k

+ 00,

Yang [xva] for each n

Blundon [xva], Groemer [viiib], Yang [xva].

In a few cases one has obtained the exact value of O n , k or a non-trivial upper estimate for O:.,. A lower estimate (greater than k ) for O:,, in the general case was derived by G. Fejes Toth [xva]. He constructed for a k-fold covering of the space by equal balls an associated "kth order Dirichlet-Voronoi tiling" of R". This tiling is a k-fold .tiling and consists of polyhedra which are star with respect to the centre of the associated ball in the covering.

xvi. Product of inhomogeneous linear forms A conjecture which is usually attributed to Minkowski (but see Dyson [49a]) can be formulated in the following way: Any lattice A in R" is a covering lattice of the star body. where P,(x)

= x1 ... x,,

for x E R"

Further, for any lattice A of the form D Y , where D is a diagonal matrix, and only for such lattices, the constant 2-" cannot be replaced by a smaller number. An arithmetical version of the conjecture reads as follows: For a given system of n real linear forms t1,.. .,&, in n variables with deter-

620

INHOMOGENEOUS FORMS

CH. 7

minant A # 0, and for any set of n real numbers C I ~ , . . . , Cthere ~, are integer values of the variables u = ( u l , . . .. u n ) such that

Here equality occurs if and only if after a suitable unimodular integer transformation of the variables the linear forms have the form B1ul,. . ., Bnun while a1 = +PI (modB1).. ., M, s $,,(mod&). This conjecture has attracted the attention of mathematicans for a long time. It remains doubtful whether it is true for all n. By the time the first edition of the book appeared the conjecture had been proved for n 5 4. These results, along with partial results (weaker estimates) for general n and related material are presented in sect. 49.2, 50, 51. The case n = 2 is discussed at length in sect. 47. In recent years the conjecture was proved for n = 5, the older asymptotic estimates were improved, and classes of lattices for which the conjecture is true were investigated in more detail. In many cases the proofs of these results are based on new ideas. In general they are rather involved, but in most cases no tools from outside the geometry of numbers are used. xvi.1. Up until now the conjecture has been verified for n 5 5. Minkowski [DA] himself proved the Conjecture for n = 2. A multitude of other proofs and various refinements of his result are reviewed in sect. 47.1 and 49.2. Most proofs for n = 3, 4, 5 consist essentially of the deduction of the following propositions for n 5 5 : (i) Given a lattice A in R", there is an ellipsoid E of the form { x : ~ ~ ( x...~ +)y ~, , (+~ , ) ~5 1) such that A is admissible for E but has n independent points on the boundary of E. (ii) Let A and E be as in (i). Then A is a covering lattice for C&&)E. That (i) and (ii) imply the truth of the conjecture is easy to see; cf. sect. 49.2. A very helpful tool for the proof of (i) is the following result of Birch and Swinnerton-Dyer [49a] which is also of interest on its own: (iii) Provided the conjecture of Minkowski holds in n - 1 dimensions,

8 xvi

PRODUCT OF INHOMOGENEOUS LINEAR FORMS

62 1

it holds in n dimensions for those lattices for which the homogeneous minimum

A(P”, A ) = inf(lP,(x)l: x E A\(o}> is equal to 0. The first proof of the conjecture for n = 3 was given by Remak [49a]. Simplified proofs are due to Davenport [49c] and Skubenko [xvib] in his treatment of the case n = 5. These proofs concentrated on the derivation of (i) and (ii). Proofs along different lines were given by Birch and Swinnerton-Dyer [49a] who applied (iii), and by Narzullaev [xvia, b, e l who used DOTU-matrices; see sect. 49.2 and subsection 3 below. The case n = 4 was first settled by Dyson [49a] who provided proofs of (i), (ii) for n = 4. Dyson’s proof of (i) borrows powerful tools from algebraic topology. An elementary but still intricate proof of the conjecture was obtained through the combined efforts of several authors: Since the conjecture holds for n = 2, 3, it is true for those lattices A in R 4 for which i(P4,A ) = 0 by virtue of Proposition (iii) of Birch and Swinnerton-Dyer [49a]. Bambah and Woods [xvia] proved (i) for lattices not having points # o in any coordinate plane and thus, in particular, for those lattices A for which A(P,, A ) > 0, see also Woods [xvid]. A different proof of this is contained in Skubenko’s [xvib] paper dealing with the case n = 5. Alternative proofs of Proposition (ii) for n = 4 are due to Hofreiter [xvia] (a dozen years before Dyson’s proof), Cleaver [xvia, b] and Woods [xvia]. The case n = 5 was mastered by the work of Skubenko [xvia, b,c] and Woods [xvib]: The latter proved (ii) for n = 5, the former succeeded in establishing (i) for lattices not having points # o in any coordinate plane. By virtue of (iii) these results imply the truth of the conjecture for n = 5. The basic idea of Skubenko’s proof consists of studying the mapping x = ( x i , .. .,x5) -,

.., ( x ~ ) ~ ) for x E R5. Under this mapping a lattice in R 5 not having any point

$ 0 on a coordinate plane is mapped onto a discrete point set in the positive orthant. The closed convex hull of this point set (excluding o) is an unbounded polyhedron with infinitely many facets. A suitable facet then leads to the ellipsoid E in (i). A clear and detailed alternative proof

622

CH. 7

INHOMOGENEOUS FORMS

of (i) which follows roughly the general line of Skubenko's proof is due to Bambah and Woods [xvic]. For n = 6 proposition (ii) was proved by Woods [xvie]. xvi.2. The prQof of the conjecture for general n seems to be exceedingly difficult but there exist (weak) approximations to it in form of asymptotic bounds. Older results in this direction are discussed in sect. 49.2. They have been improved in recent years. As in sect. 49.2, let po(P,) be the greatest lower bound of the numbers p > 0 having the property that any lattice A is a covering lattice of the star body

The following upper bounds for p o ( P , ) have been obtained : 2

-:.

2 -h,-',CI, -+ 2e - 1 = 4.436563 . . . 2-iflp;',/3,, 2(2e-1) = 8.873127. . . 2-tny;',yn + 3.0001(2e-l) = 13.310134... 2-fnd;1,6, -+ 3(2e-1) = 13.309690... 2-tne2n-flog+n for large n e2 = 7.389056.. . 2-ine25.6 n -'TIogSn for large IZ e25.6= 1.312014..: 10" 2 - i n n - ~ l o g ~ 5.95323.. ". . for large n --f

cebotarev [49a] Davenport [49e] Woods [49a] Bombieri [49b] Gruber [49a] Skubenko [xvid, e l Narzullaev and Skubenko [xvib] Muhsinov [xvib]

An even more refined results of this type has been proved by Andrijasjan, Il'in and MalySev [xvia]. For other results in this direction see Muhsinov [xvic, d]. cebotarev's result was reproved by Narzullaev [xvic]. The proofs of the above estimates are elaborations of the methods presented in sect. 49.2 : One applies Minkowski's fundamental theorem or Blichfeldt's theorem to o-symmetric compact convex bodies or other sets related to the star body {x: IP,(x)l S l } or translates of it.

xvi.3. The conjecture is true for certain classes of lattices. One such class is described in a theorem of Schneider [49a] stated in sect. 49.2.

5 xvi

623

PRODUCT OF INHOMOGENEOUS LINEAR FORMS

Schneider's result was reproved by Mordell [xvia] who applied the method of the additional variable (see sect. 47.3, 51.3). A second more important class is connected with so-called DOTUmatrices, first introduced by Macbeath [49a]. A (real) n x n matrix A is a DOTU-matrix if it can be represented as the product of a diagonal matrix, an orthogonal matrix, an upper triangular matrix with 1's in the diagonal, and an integral unimodular matrix. If A is a DOTU-matrix it is easy to show that the Minkowski conjecture holds for the lattice A Y , see Theorem 49.3 due to Macbeath [49a]. Macbeath [49a] proved that each 2 x 2 matrix is DOTU and Narzullaev [xvia, b, el established an analogous result for 3 x 3 matrices. Let the space of n x n matrices be endowed with its usual topology (which is introduced by identifying n x n matrices with points of R"' or by one of the common matrix norms, such as the one specified in sect. 17.1). The set of DOTU-matrices is dense in the space of all n x n matrices as each n x n matrix with rational entries is DOTU. Macbeath [49a] showed that this set is also open. A related result is contained in Narzullaev [xvif]. In [xvig] Narzullaev gave a computational criterion for a matrix to be DOTU. Gruber [xvic, d] and Ahmedov [xvia, b, c] proved the existence of square matrices which are not DOTU. The proofs can be described as follows: if A is a DOTU-matrix then

On the other hand one can find matrices A for which this inequality is not satisfied. It suffices to find a totally real algebraic number field F of degree n such that the discriminant of F (see sect. 4.1) satisfies the inequality A , < n". Then the matrix formed by the elements of a basis for the integer elements of F together with the algebraic conjugates of these numbers provides an example. The existence of such fields is guaranteed by results of class field theory, in particular by the existence of infinite class field towers. Skubenko [xvif] gave an example for n = 2 8 8 0 : L e t q l =2cos(2n/m),wherern=3.5.7.11.13= 15015.The conjugates of q l (including q l ) in the field Q(ql) are the 2880 different numbers among

2n1 2cos--, m

where 1 = 1,2,... and

(I,rn)

=

1.

624

CH. 7

INHOMOGENEOUS FORMS

I 1

Denote these by q l , . . ., qn. Then

lqlq:. . .qy-' 1q,q;. . . q;-

............... 1qns:...q;-1

is not a DOTU-matrix. An example for n Lenstra [xvia]. Lenstra took the field F = Q

=

64 was communicated by

( :3k r , , 2cos--,

5 + 2 3

which is totally real of degree 64, whilst the discriminant of F satisfies =

56.96 ... < 64.

If a square matrix A is not a DOTU-matrix, a remark of Ahmedov says that the 'enlarged' matrix

is also not DOTU. There are results on the measure of the set of lattices for which the conjecture holds. All these results are due to Gruber [xvia]. We first consider the measure on the space of lattices of determinant 1 introduced by Siege1 [19a] and described in sect. 19.3. (This measure essentially is the restriction of a Haar measure.) Assume that the measure is normalized such that the whole space has measure 1. For all sufficiently large n each lattice A from a set of lattices of determinant 1 with measure L 1- e - 0 . 2 7 9 n is a covering lattice of the star body

(Note that by Stirling's formula -

e-"J2n

n! nn - 1

N

fi

as n

+

co.)

9: xvi

PRODUCT OF INHOMOGENEOUS LINEAR FORMS

625

In the proof of this result it is shown by means of a density argument that each such lattice A is a covering lattice of a compact convex body contained in the star body described in (1). The proof makes essential use of a measure theoretic result of Schmidt [19g]. A measure on the space of all n x n matrices is defined by simply taking Lebesgue measure or R"'. (This measure is not a Haar measure on the group of non-singular n x n matrices.) The following results are formulated using this measure but, of course, it would have been possible to state them also by using the measure introduced by Siegel. Assume that Minkowski's conjecture holds in dimensions 2, 3, . . ., n - 1. Then for almost all n x n matrices A it holds for the lattice A Y . The proof of this result is based on the theorem of Birch and Swinnerton-Dyer cited in subsection 1 and a measure-theoretic result of Schmidt [ 19il. As a special case of a slightly more general asymmetric result we state the following 2-dimensional result, the proof of which again makes use of a theorem of Schmidt [19i]. Let p > 4 be chosen. For almost all 2 x 2 matrices A the lattice A Y covers the plane infinitely often by the star body

The set of matrices A for which A Y is not a covering lattice of the star body (2) is closed and nowhere dense in the space of all 2 x 2 matrices. Whether these results hold for smaller values of p is not known. A result of Ennola [47a] says that p may not be taken to be less than 1/(16+6&) = 1/30.696938 ... > Measure-theoretic results of a completely different type but related to the conjecture were given by Bombieri [49c]. The foregoing shows that the conjecture holds in any dimension for a very large set of lattices which is dense in the space of all lattices. Unfortunately this does not allow us to conclude that the conjecture holds for all lattices, because the inhomogeneous minimum p(P,,,A ) is an upper semicontinuous but not necessarily continuous functional of A, see Gruber [ixa] and sect. ix.2.

A.

xvi.4. In this subsection we shall collect various results related to the conjecture on the product of inhomogeneous linear forms : Results

626

INHOMOGENEOUS FORMS

CH.

7

depending on the homogeneous minimum A(P,,, A ) , an attack of Bombieri, and asymmetric results. Proposition (iii) of Birch and Swinnerton-Dyer [49a] is a first example of results related to Minkowski's conjecture in which the homogeneous minimum A(P,,, A ) is taken into account. Gruber [xvic] proved that for n = 3 each lattice A is a covering lattice for the star body

{

d(A)I

x : IPJ(X)l 5 (1 - M A ) - 23

'

where = 1.2016 ... and A = A ( P 3 , A ) / d ( A )The . proof is based on the fact that each 3 x 3 matrix is DOTU. A similar result holds for arbitrary n if A has the form A Y , where A is a DOTU-matrix. Results of the same type but for arbitrary n and any lattice A are due to Bakiev, Pen and Skubenko [xvia], Bakiev [xvia] and Skubenko and Bakiev [xvia]. In order to present still another line of attack we remind the reader of the definition of p ( P n , A , u ) = inf(lP,(x-a)l:xeA),

for a lattice A and for a E R". It was proved by Bombieri [xvia, 49c] that for any lattice A , any point a and for any E > 0: p(P,, A, ka) S 2-"d(A) for a suitable positive integer k S 2f@+l), p(P,, A, ka) 5 cd(A) for a suitable positive integer k S 1 + E - ' .

Refinements of Bombieri's results are due to Gruber [xvia] and Narzullaev [xvid]. In connection with the conjecture on the product of inhomogeneous linear forms many results involving asymmetric inequalities have been proved, see sect. 50 for a review. Let k = 0,1,. . ., n. Gruber [xvie] conjectured that any lattice A is a covering lattice of the set

The cases k = 0 , l were settled by Chalk [50a], [51a] (see also Narzullaev [xvic] and Cole [SOa], respectively). Minkowski's conjecture

Q xvii

INHOMOGENEOUS FORMS

621

corresponds to the case k = n. For n = 3 the case k = 2 which is not covered by either of the results of Chalk, Cole and Remak [49a] was settled by Bambah and Woods [xvib]. Thus in dimension n = 3 the conjecture holds generally. For various other asymmetric results we refer to the articles of Grover [xvia], Gruber [xvia] and Woods [xvic]. Among papers related to the product of inhomogeneous linear forms not mentioned in the foregoing are Gruber [mib], Muhsinov [xvia,e] and Narzullaev and Skubenko [xvia].

xvii. Inhomogeneous forms We report briefly on new results, mainly for indefinite quadratic forms. The results concern symmetric as well as asymmetric and onesided inequalities. Older results will be included in the exposition. xvii.1. Let Q be a non-degenerate indefinite quadratic form in n variables, n

with discriminant D = det(sij) and of type ( r , s ) , i.e., Q can be written - . . . where r + s = n and in the form ( Y ~ ) ~ ++. .( .~ , ) ~ - ( y ~ + ~ ) ~-(y,)', y , , . . ., y, are independent linear forms in x l r . . ., x,. The difference r - s is the signature of Q. There exists a positive constant p such that for each point a E R" the inequality IQ(u -a)l 5 (p1Dl)"" has a solution u E Y for each form Q of given type ( I , s); see sect. 49.4. There even exists a positive number v such that the inequality 0 < Q(u-a) S (vlDl)'/" is always solvable (Blaney [~OC], Foster [xviia]). Given the type ( I , s) we denote the infima of the numbers p, v having the properties stated by p r ,s, vr, ~,respectively. These quantities can be interpreted as absolute inhomogeneous minima ; more precisely, p:,'; and v:,'; are the symmetric and the one-sided absolute inhomogeneous minimum of quadratic forms of type (r, s) with discriminant f 1; cf. sect. 46. Cleaily, p r , s = p s , r . In (1) we list explicit values for p r , s for small values of r and s. We also include older results reported in sect. 47 and 49.

628

INHOMOGENEOUS FORMS

CH. 7

Minkowski [DA] Davenport [49h] Dumir [xviia] Birch [49a] Hans-Gill and Raka [xviia, b] p1.5

Raka [xviia]

=

pr,s = 1 11 1 4, 4, 3 , 2 r

1 for n L 21 according as r - s = 0, k l , i 2 , ?3,4(mod8)

Watson [xviia]

The proofs of the results for p l , l , . . ., p l , are arithmetical in character; they are based on induction, separation of binary forms, and asymmetric results for forms in fewer variables, in particular, binary forms. Watson conjectured that his results are valid for n L 4 ; the above result for p l , l , . . ., pl, are in accordance with this conjecture. Watson's treatment was based on a reduction of the problem to the case of integral forms and a careful analysis of that case; the analysis comes down to a problem concerning the distribution of quadratic residues (Watson [xviib]). Related asymmetric results were given by Watson [xviic], Jackson [xviia], and Bambah et al. [xviia]. Further support for Watson's conjecture was found by Raka [xviib, c] who treated the cases r - s = f t , f2,f 3 , +4. The results for vr.s are collected in the following list v1.1 v1.2

4 =8 =

16 v2.1 = 4 v 2 , 2 = 16 and v3, = 9 v 3 , 2 = 16 and v4, =8 vr,s = 2"/(r-s+ 1) for r --s ~ 1 . == 3

Vr,r+l=

An estimate for

for r = 2, forr 2 3

= 0,1,2,3

sect. 50.1 Dumir [xviib] Dumir and Hans-Gill [xviib] Barnes [50a], Blaney [50d] Dumir [xviic] Hans-Gill and Raka [xviic, d] Bambah et al. [xviia, b] Bambah et al. [xviic]

is given in Raka [xviia]. As in the case of

~ 1 . 4

5 xviii

GEOMETRY OF NUMBERS IN OTHER SPACES

629

symmetric minima the proofs of the above results are of an arithmetical character. They are complicated in structure and partly based on results for asymmetric inequalities. Zero forms are treated separately. In many cases the ‘critical’ forms were found; these forms are (multiples of) are isolated. The former rational forms. The minima v 1 , 2 and result is due to Dumir and Hans-Gill; it is not yet published. For the latter see Dumir and Hans-Gill [xviib] and Bambah et al. [xviic]. The minimum v l , is not isolated; see sect. 50.1. Asymmetric inequalities were studied for their own sake or for purposes of applications by Blaney [50a, c], Dumir [xviic], Hans-Gill [xviia] and Hans-Gill and Raka [xviiie, f, g]. See also Foster [xviia], Gruber [xvia], Jackson [xviia] and Bambah et al. [xviia]. In the case n = 2, more explicit results have been obtained for particular forms. The inhomogeneous minimum of an indefinite (binary) quadratic form Q is denoted by p ( Q , Y ) ; the supremum of p ( Q , Y)’/ID(Q)I is the minimum pl,1 , see sect. 46 and subsection 1. Varnavides [xviia, b] determined p ( Q , Y ) for some sequences of binary indefinite quadratic forms (see also sect. 47.4), Godwin [xviia] proved that ( x ~- 2) 3~( ~ , ) ~ has a second inhomogeneous minimum and that this minimum is not isolated, and Blanksby [xviia] studied a restricted inhomogeneous minimum. For norm forms of quadratic number fields there is a connection with the problem whether the Euclidean algorithm holds in the corresponding field; see sect. 47.5. For a historical survey of the question as well as an extension in which second homogeneous minima play a role the reader is referred to Van der Linden [xviia]. xvii.2. Other forms for which the inhomogeneous minimum was considered are norm forms of totally real algebraic number fields (see Godwin [xviib], Van der Linden [xviia] and the references given by the latter), determinants of symmetric matrices whose elements are linear forms (Davenport [xviia]), and distance functions of planar star-shaped domains; see sect. 47.7 and Dumir and Hans-Gill [xviic].

xviii. Geometry of numbers in other spaces

For a long time people have tried to develop analogues of the geometry of numbers in other spaces. Above all attention has been

630

INHOMOGENEOUS FORMS

CH. 7

paid to finite-dimensional vector spaces over other fields than the field of reals R . The fields considered are of various types and include algebraic number fields (possibly extended to algebras over R ) , nonArchimedean fields and so-called global fields. In particular, analogues have been derived of the fundamental theorems of Blichfeldt and Minkowski and Mahler's compactness theorem. The results have applications to algebraic function fields. Part of the results were extended to topological spaces on which operate transitive groups of transformations. In this section we make some brief remarks on the subject. xviii.1. In this subsection we consider generalizations and analogues of fundamental results of the geometry of numbers. All vector spaces are of finite dimension. K is an algebraic number field. Contributions to the geometry of numbers over fields other than R were given by Minkowski [DA, Ch. 61 (vector spaces over imaginary quadratic fields), Weyl [IlOa], K. Rogers and Swinnerton-Dyer [xviiia] and Chalk [xviiia] (vector spaces over any field K ) , Mahler [xviiia] and Lutz CAP] (vector spaces over a field of formal Laurent series, with coefficients in a field K ) , and Cantor [xviiia] and McFeat [xviiia] (vector spaces over a field of adeles, i.e., valuation vectors of a field K ) . A survey is given by Chalk [xviiie]. The quoted authors gave analogues of the fundamental theorems of Minkowski. To a certain extent, proofs run along similar lines to the classical case. But there are at least formal differences. Linear dependence of vectors, linear transformations, lattices, and successive minima with respect to a convex body have to be defined appropriately in the various cases. For spaces over fields of Laurent series Mahler [xviiib] found that a bounded convex body is necessarily a parallelepiped and that the inequality corresponding to Minkowski's second theorem (on successive minima) is in fact an equality. In the case of a vector space A" over a field of adeles (a global field) the space or groups of linear transformations of A" are defined as (restricted) infinite direct products. Groups of linear transformations of A" can be topologized in various natural ways; for the set of lattices in A", which can be identified with a quotient group of such a (multiplicative) group of linear transformations, McFeat [xviiia] derives an analogue of Mahler's compactness theorem. Mahler's compactness theorem, as well as Blichfeldt's theorem, can be extended to general classes of topological spaces. For Mahler's

§ xviii

GEOMETRY OF NUMBERS IN OTHER SPACES

63 1

theorem this was done by Chabauty [17a] and Macbeath and Swierczkowsky [xviiia], for Blichfeldt’s theorem by Santalo [xviiia ; IG, Sect. 11.10.61 and in a special case by Tsuji [6a,xviiia]. The spaces considered are topological groups or spaces on which operate transitive groups of transformations; they can be partitioned into sets of fundamental domains with respect to discrete subgroups of the groups in question. We note that, from an abstract point of view Blichfeldt’s theorem is based on an interchange of the order of integration in an appropriate integral expression ; see sect. 6 and Chabauty [xviiia]. The result of Mahler [xviiia] for parallelepipeds in a space over a field of Laurent series was applied by Armitage [xviiia]. The latter took an algebraic extension of such a field and showed that Mahler’s result on successive minima (with respect to the extended field) is essentially the Riemann-Roch theorem for an arbitrary field of algebraic functions in one variable. xviii.2. Analogues of other results in the geometry of numbers were given for spaces over complex fields by Minkowski [DA, Ch. 601, Mulholland rxviiia], Saffari [xviiia], Feit [xviiia], Hansen [xviiia], Karian [xviiia] and Quebbemann [xid], for spaces over fields of Laurent series or adeles by Mahler [xviiib], Armitage [xviiib], Jarnik [xviiia], Dubois [xviiia], Aggarwal [xviiia, b, c], Cusick [xviiia]. The results concern lattice constants, products of homogeneous or inhomogeneous linear forms, classification of unimodular lattices (sect. xi), Markov spectrum and covering problems in the case of complex fields; anomaly of a convex body (sect. 18), products of linear forms, transfer theorems and problems for inhomogeneous linear forms in the case of fields of Laurent series. Armitage [xviiic] relates a conjecture on the minimum of the products of n linear forms to the Riemann hypothesis in function fields (see MR50 12923). The Markov phenomenon (sect. 43.4 and xiv.5) was studied in connection with Fuchsian groups and LobaEevskian geometry by A. L. Schmidt [xviiia, b, c] and GorSkov [xiva], respectively. Boroczky [xviiia] considered packings of balls in hyperbolic space, see also Coxeter [xviiia].