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

TL;DR

This paper proves that for lacunary sequences of positive real numbers, all higher-order spacing correlations are Poissonian when scaled by almost all real numbers, extending previous results on pair correlation.

Contribution

It establishes that higher correlations and the nearest-neighbour spacing distribution are Poissonian for real-valued lacunary sequences, generalizing earlier work on pair correlation.

Findings

Higher correlations are Poissonian for lacunary sequences.

Nearest-neighbour spacing distribution is Poissonian.

Extends previous pair correlation results to all higher correlations.

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.