Poissonian pair correlation for higher dimensional real sequences

Abstract

In this article, we examine the Poissonian pair correlation (PPC) statistic for higher-dimensional real sequences. Specifically, we demonstrate that for $d\ge 3$, almost all $(\hat{I}{\pm }_1,\dots ,\hat{I}{\pm }_d)\in {\mathbb{R}}^d$, the sequence $(\{x_n\hat{I}{\pm }_1\},\dots ,\{x_n\hat{I}{\pm }_d\})$ in $[0,1{)}^d$ has PPC conditionally on the additive energy bound of $(x_n).$ This bound is more relaxed compared to the additive energy bound for one dimension as discussed in [1]. More generally, we derive the PPC for $(\{x_n^{(1)}\hat{I}{\pm }_1\},\dots ,\{x_n^{(d)}\hat{I}{\pm }_d\})\in [0,1{)}^d$ for almost all $(\hat{I}{\pm }_1,\dots ,\hat{I}{\pm }_d)\in {\mathbb{R}}^d.$ As a consequence we establish the metric PPC for $(n^{\hat{I}{¸}_1},\dots ,n^{\hat{I}{¸}_d})$ provided that all of the $\hat{I}{¸}_i$'s are greater than two.