On higher dimensional Poissonian pair correlation

In this article we study the pair correlation statistic for higher dimensional sequences. We show that for any $d\ge 2$, strictly increasing sequences $(a_n^{(1)}),\dots ,(a_n^{(d)})$ 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 $(\{n\alpha \},\{[n{\log }^{A}n]\beta \})$ ($A\in [1,2]$) 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 Kaltenböck [15]. The proof uses estimates for ‘Generalized’ GCD-sums.