On higher dimensional Poissonian pair correlation

TL;DR

This paper investigates higher-dimensional Poissonian pair correlation for sequences, establishing conditions under which sequences exhibit this property and providing counterexamples to previous conjectures, using GCD-sum estimates.

Contribution

It proves that sequences with certain additive energy bounds have metric Poissonian pair correlation in higher dimensions, and shows specific sequences do not have this property, answering prior open questions.

Findings

Sequences with $O(N^{3- ext{delta}})$ additive energy have Poissonian pair correlation.

Certain two-dimensional sequences exhibit Poissonian pair correlation for almost all parameters.

Counterexamples to previous conjectures are provided using GCD-sum estimates.

Abstract

In this article we study the pair correlation statistic for higher dimensional sequences. We show that for any $d\geq 2$, strictly increasing sequences $(a_n^{(1)}),\ldots, (a_n^{(d)})$ of natural numbers have metric Poissonian pair correlation with respect to sup-norm if their joint additive energy is $O(N^{3-\delta})$ for any $\delta>0$. Further, in two dimension, we establish an analogous result with respect to $2$-norm. As a consequence, it follows that $(\{n\alpha\}, \{n^2\beta\})$ and $(\{n\alpha\}, \{[n\log^An]\beta\})$ ($A \in [1,2]$) have Poissonian pair correlation for almost all $(\alpha,\beta)\in \mathbb{R}^2$ with respect to sup-norm and $2$-norm. This gives a negative answer to the question raised by Hofer and Kaltenb\"ock [15]. The proof uses estimates for 'Generalized' GCD-sums.