A pair correlation problem, and counting lattice points with the zeta function

TL;DR

This paper extends the theory of pair correlations to real-valued sequences, establishing criteria for Poissonian behavior and proving that sequences of the form (n^θ α) exhibit this property for almost all α when θ > 1.

Contribution

It develops a new framework for analyzing pair correlations of real-valued sequences, generalizing previous integer-based results and applying it to sequences like (n^θ α).

Findings

Sequences (n^θ α) have Poissonian pair correlation for almost all α when θ > 1.

The framework applies to sequences of real numbers, broadening the scope of pair correlation analysis.

Provides criteria linking additive energy of sequences to Poissonian behavior.

Abstract

The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form $(a_n \alpha)_{n \geq 1}$ has been pioneered by Rudnick, Sarnak and Zaharescu. Here $\alpha$ is a real parameter, and $(a_n)_{n \geq 1}$ is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number $\alpha$, in terms of the additive energy of the integer sequence $(a_n)_{n \geq 1}$. In the present paper we develop a similar framework for the case when $(a_n)_{n \geq 1}$ is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number $\theta>1$, the sequence $(n^\theta \alpha)_{n \geq 1}$ has Poissonian pair correlation for almost all $\alpha \in \mathbb{R}$.