Remarks about inhomogeneous pair correlations

TL;DR

This paper introduces an inhomogeneous generalization of pair correlations and explores their behavior in a doubly metric setting, extending existing results and establishing new findings in the distribution of fractional parts.

Contribution

It defines a new inhomogeneous pair correlation concept and analyzes its properties in a doubly metric framework, providing novel results beyond previous homogeneous studies.

Findings

Many results from the homogeneous setting carry over to the inhomogeneous case.

The doubly metric framework allows establishing new results not available in singly metric scenarios.

The paper advances understanding of the distribution of fractional parts in inhomogeneous settings.

Abstract

Given an infinite subset $\mathcal A \subseteq\mathbb N$, let $A$ denote its smallest $N$ elements. There is a rich and growing literature on the question of whether for typical $\alpha\in[0,1]$, the pair correlations of the set $\alpha A \pmod 1\subset [0,1]$ are asymptotically Poissonian as $N$ increases. We define an inhomogeneous generalization of the concept of pair correlation, and we consider the corresponding doubly metric question. Many of the results from the usual setting carry over to this new setting. Moreover, the double metricity allows us to establish some new results whose singly metric analogues are missing from the literature.