The distribution of spacings of real-valued lacunary sequences modulo one

Abstract

Let ${\left(a_n\right)}_{n=1}^{\infty }$ be a lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all $\alpha \in \mathbb{R}$, the pair correlation of ${\left(\alpha a_n\right)}_{n=1}^{\infty }$ mod 1 is Poissonian. We show that all higher correlations and hence the nearest-neighbour spacing distribution are Poissonian as well, thereby extending a result of Rudnick and Zaharescu to real-valued sequences.

12 pages