40. Uniform Distribution Modulo 1

40.

UNIFORM DISTRIBUTION MODULO

1

197

In alternate intervals [T,,, T,,,,] on the right half-axis, the function - g ( r ) vanishes, and in the remaining intervals If(r) -g(t)l = 1 on a subset whose measure is approximately half the length T,,, - T,,. A similar comment is valid for the left half-axis. If the difference f(r) - g(t) had an autocorrelation, the limit as T approached infinity of

f(r)

would exist, but if the Tn are widely enough spaced, say T,, = 4(n! + l), the function Q ( T ) oscillates between 0 and $.

40. Uniform Distribution Modulo 1 A real number x may be written in a unique way in the form x = [XI + (x) where [x] is an integer and (x) is in the interval 0 5 x < 1 ; the number (x) is the representative of x modulo 1. Given a sequence ak of real numbers, we study the sequence modulo 1, that is, the sequence (ak) in [0, 1). The sequence is said to be uniformly distributed mod 1 if, for every interval I contained in [0, I), the proportion of (ak)which falls in I is asymptotically equal to the length of I. More formally: if N ( m , I) is the number of (ak) with k 5 m which are in the interval I , then m

m

exists and equals the Lebesgue measure of I. It is also possible to think of the uniform distribution in another way: the first m numbers (ak) determine a measure in the unit interval consisting of point masses I/m at the m (not neces sarily distinct) points (a&; this sequence of measures pm consists of measures of total mass 1, and by Helly’s theorem has at least one weakly convergent subsequence, converging to a limit measure p. The sequence is uniformly distributed mod 1 if and only if p is the Lebesgue measure, and in this case it was not necessary to pass to a subsequence. This remark is virtually a proof of the following theorem, which we nevertheless prove without invoking Helly’s theorem.

Theorem: The sequence 0, is uniformly distributed mod 1 if and only if for every Riemann integrable functionf(x), periodic with period 1, the limit l N 1 lim - f ( a k ) exists and equals f ( x ) dx ~

+

1

Nk=i

m

lo

198

111.

HARMONIC ANALYSIS

PROOF: If the limit exists, as asserted in the theorem, we take forf(x) the characteristic function of the interval Iextended over the axis with period 1 to infer that lim, N(m, I)/m exists and equals the Lebesgue measure of I, that is, that the sequence is uniformly distributed mod 1. On the other hand, if the sequence is uniformly distributed mod 1, the assertion of the theorem holds for any functionf(x) which is a finite linear combination of such characteristic functions of intervals extended by periodicity with period 1. Now for any function f ( x ) , Riemann integrable in the interval, there exist two finite linear combinations of characteristic functions of intervals h(x) and g(x) such that g(x) S f ( x ) 5 h(x) and h(x) - g(x) < E ; extending those two functions by periodicity we have

and the limits at either end of this inequality differ by at most E. Thus, the theorem is proved, and from it we obtain a criterion established by H. Weyl.

Theorem (Weyl): The sequence uk is uniformly distributed mod 1 if and only if, for every integer I > 0,

I N

lim N

1 eiZnfak exists and is

Nk=i

0.

PROOF: If the sequence uk is uniformly distributed mod 1, we invoke the noting that l o l f ( x )dx = 0. On previous theorem for the functionf(x) = eiZnfx, the other hand, if the limits considered in the theorem exist and are 0, then for A,,, eilnmx, every trigonometric polynomial P ( x ) =

zm

1

lirn N

1N P(ak)

Nk=i

exists and equals

Iff(x) is the characteristic function of an interval I i n [0, 1) extended periodically with period 1, there exist two trigonometric polynomials P ( x ) and Q(x) so that P ( x ) S f ( x ) S Q(x) and Q(x) - P ( x ) < E ; we infer that lim( l/N)C;= f(ak) exists and equals the length of I, hence, that the sequence is uniformly distributed mod 1.

40.

UNIFORM DISTRIBUTION MODULO

1

199

Let 2 be an irrational number and a, be the sequence ak = k2, k 2 1; this sequence is uniformly distributed mod 1, since for every 1 > 0,

and this quantity is bounded in absolute value by 2/Nlsin(2nlA)l and hence converges too. Had 1 been rational, of course only a finite set of residues mod 1 would occur. We pass to a theorem of van der Corput.

Theorem (van der Corput): Let the sequence ak have the property that for every integer h > 0 the sequence ak+h- ak = bk is uniformly distributed mod 1 ; then this also holds for the sequence ak .

PROOF: For a fixed integer 1 > 0, define the function f ( t ) equal to 0 for t < 0 and equal to eiZnLak in the interval k - 1 S t < k. This function has an autocorrelation, since the functions (1/2T)(fT * f T ) ( s )which are linear in intervals of the form ( k - 1, k), converge for integral values of s. This convergence is obvious for s = 0, while for s = h > 0 and larger integer values of T,

which converges to 0 by hypothesis. The autocorrelation therefore exists and is a triangle function: it vanishes for JsI 2 1 and is equal to +(l - Isl) for Is1 < 1 ; its Fourier transform is an absolutely continuous measure p which has therefore no mass at the origin. From the van der Corput theorem, then,

and this means that the numbers ( 1 / 2 N ) ~ ~eiZnfak = , converge to 0 with increasing N . Since I was arbitrary, it follows that the ak are uniformly distributed mod 1.

Corollary:

Let the polynomial P ( x ) = A,xm

+ A,,,-,x"'-~+

* - *

+ A , x + A0

have an irrational leading coefficient A,,,; then the sequence ak = P ( k ) is uniformly distributed mod 1.

200

111. HARMONIC ANALYSIS

PROOF: We argue by induction; for m = 1, the theorem has already been shown. For larger m and any integer h > 0, the polynomial Q(x) = P ( x + h) - P ( x ) is of lower degree and has an irrational number as its leading coefficient, and so Q(k)is uniformly distributed mod 1. The h being arbitrary, the previous theorem guarantees that ak is uniformly distributed mod 1. It is not difficult to extend the criteria of the previous theorems to sequences of points in R"; these sequences are reduced mod 1 to sequences of representative points in the unit cube of R", each coordinate being taken mod 1 separately. The most interesting case occurs when n = 2, where the sequence of points has the coordinates ( a k ,bk),the representatives mod 1 being ((a&, (bk)) in the unit square. The sequence is uniformly distributed in the square if and only if for every pair of integers ( I , h) not both 0, the sums

converge to 0. If we consider a point moving with uniform velocity in the x , y plane along a linear path of slope m, the coordinates of the point may be written as functions of time: ~ ( t =) 1, y ( t ) = mt + b, and when these coordinates are reduced mod 1, we obtain a family of lines of slope m in the unit square. As the time t runs through the positive integers, we obtain a set of points in the square

Fig. 8.

4 I.

SCHOENBERG'S THEOREM

201

which is uniformly distributed there, provided that the slope m is irrational, since -

CN

ei2n(l+hm)k

Nk= I

ei l n h b

converges to 0 with increasing N because (I + hm) is irrational if m is irrational. If we make the further reduction shown by Fig. 8

W )= min C(x(", f - (x(r))l, y o ) = min f - (m19 C(Y(N9

we obtain a continuous path in the square of side length f which is that of a billiard ball on a square billiard table, the ball being reflected by the sides of the table in the usual way. Thus, the slope being irrational, the ball spends equal amounts of time in equal areas of the table. Note that the initial condition, essentially the coordinate b, has nothing to do with the long term behavior of the ball. When the slope m is rational, the path of the ball is periodic.

41. Schoenberg's Theorem The measure w on R" which consists of a uniform distribution of unit mass on the surface 1x1 = 1 clearly plays an important role in the study of functions and distributions which are spherically symmetric, that is, are invariant under the orthogonal group. Hence, it is natural to expect that the Fourier transform of that measure will appear in a variety of applications and will be a particularly important function of positive type. We study that function in this section, but find it convenient to normalize the measure differently, and to consider the measure w, d o ; we recall that w, = 2 d 2 / r ( n / 2 ) . Let

G,(t)

= (2n)-"'2je-"x0wn dw(x) ;

this is evidently a function of positive type, and since the support of the measure is compact, it can be extended to an analytic function of n complex variables. Since the measure is invariant under the transformations of the orthogonal group, so is the function G,(t), which is therefore a function of radius alone, and we may write = Gn(O>fn(

It I)

7