Poissonian pair correlation for higher dimensional real sequences

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 $({\alpha }_1,\dots ,{\alpha }_d)\in {\mathbb{R}}^d$, the sequence $(\{x_n{\alpha }_1\},\dots ,\{x_n{\alpha }_d\})$ in $[0,1{)}^d$ has PPC conditionally on the additive energy bound of $(x_n)\mathrm{.}$ 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 $(\{x_n^{(1)}{\alpha }_1\},\dots ,\{x_n^{(d)}{\alpha }_d\})\in [0,1{)}^d$ for almost all $({\alpha }_1,\dots ,{\alpha }_d)\in {\mathbb{R}}^d\mathrm{.}$ As a consequence we establish the metric PPC for $(n^{{\theta }_1},\dots ,n^{{\theta }_d})$ provided that all of the ${\theta }_i$’s are greater than two.