The distribution of spacings between fractional parts of lacunary sequences

Abstract

Given an increasing sequence of integers a(n), it is known (due to Weyl) that for almost all reals t, the fractional parts of the dilated sequence t*a(n) are uniformly distributed in the unit interval. Some effort has been made recently to understand the distribution of the spacings between elements of such sequences (normalized to have unit mean), especially for sequences of polynomials. In this note, we consider lacunary sequences, such as the sequence of of powers of 2 (that is a(n)= 2^n). We show that for almost every t, all local correlation functions of the sequence t*2^n are those of a random sequence (the Poisson model). Among other things, this implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution.