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 $$, almost all $$, the sequence $$ in $$ has PPC conditionally on the additive energy bound of $$. This bound is more relaxed compared to the additive energy bound for one dimension as discussed in [Aistleitner, El-Baz, and Munsch, Geom. Funct. Anal. 31 (2021), 483–512]. More generally, we derive the PPC for $$ for almost all $$. As a consequence we establish the metric PPC for $$ provided that all of the $$ are greater than two.