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

Abstract

Let ( a n ) n = 1 ∞ $(a_{n})_{n=1}^{\infty }$ be a lacunary sequence of positive real numbers. Rudnick and Technau showed that for almost all α ∈ R $\alpha \in \mathbb {R}$ , the pair correlation of ( α a n ) n = 1 ∞ $(\alpha a_{n})_{n=1}^{\infty }$ 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.