On Weyl's Criterion for Uniform Distribution

1. In his famous memoir [1] of 1916, Weyl gave a necessary and sufficient condition for a sequence s1, s2 , • • • of real numbers to be uniformly distributed modulo 1, namely that for each integer m # 0, It is natural to ask : what condition on N S(N)=NI] e(msn)-0 n=1 as N-. (Here e(a) = ez7ria .) This criterion has been fundamental for much subsequent work on Diophantine approximation. Now suppose that the sequence sn is replaced by a sequence s n(x) depending on a real parameter x, each s n (x) being bounded and integrable for a < x < b. Let N S(N, x) = 1 ri e(ms n(x)). will ensure that the sequence s n (x) is uniformly distributed modulo 1 for almost all x, in the sense of Lebesgue measure? We answer this question in the following theorem. THEOREM. If the series Z' N-1 I(N) converges for each integer m 0, then the sequence s n(x) is uniformly distributed modulo 1 for almost all x in a < x < b. On the other hand, given any increasing function (D(M)-which tends to infinity with M (however slowly), there exists a sequence s n (x) which is not uniformly distributed modulo 1 _for any x, and which satisfies the inequality M N-1 I(N) < <D (M). N=1 2. The proof of the first half of the theorem is based on a principle of interpolation which was used in a particular case by Weyl himself [1 : Section 7] .