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 $d\geq 3$, almost all $(\alpha_1,\ldots,\alpha_d) \in \mathbb{R}^d$, the sequence $\big(\{x_n\alpha_1\},\dots,\{x_n\alpha_d\}\big)$ 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 $\big(\{x_n^{(1)}\alpha_1\},\dots,\{x_n^{(d)}\alpha_d\}\big) \in [0,1)^d$ for almost all $(\alpha_1,\ldots,\alpha_d) \in \mathbb{R}^d.$ As a consequence we establish the metric PPC for $(n^{\theta_1},\ldots,n^{\theta_d})$ provided that all of the $\theta_i$'s are greater than two.