In this article we study the pair correlation statistic for higher
dimensional sequences. We show that for any d≥2, strictly increasing
sequences (an(1)),…,(an(d)) of natural numbers have metric
Poissonian pair correlation with respect to sup-norm if their joint additive
energy is O(N3−δ) for any δ>0. Further, in two dimension, we
establish an analogous result with respect to 2-norm. As a consequence, it
follows that ({nα},{n2β}) and ({nα},{[nlogAn]β}) (A∈[1,2]) have Poissonian pair correlation for
almost all (α,β)∈R2 with respect to sup-norm and
2-norm. This gives a negative answer to the question raised by Hofer and
Kaltenb\"ock [15]. The proof uses estimates for 'Generalized' GCD-sums.Comment: Added references and corrected typos. To appear in Journal of
Mathematical Analysis and Application