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 >= 2, strictly increasing sequences (a(n)((1))), . . . , (a(n)((d))) of natural numbers have metric Poissonian pair correlationwith respect to supnorm if their joint additive energyis O(N3-delta) for any delta > 0. Further, in dimension two, we establish an analogous result with respect to the 2-norm. As a consequence, it follows that ({n alpha}, {n(2)beta}) and ({n alpha}, {[n log(A) n]beta})(A is an element of[1, 2]) have Poissonian pair correlation for almost all (alpha, beta) is an element of R-2 with respect to sup-norm and 2-norm. This gives a negative answer to the question raised by Hofer and Kaltenbock [15]. The proof uses estimates for 'Generalized' GCD-sums. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

In this article, the pair correlation statistic for higher dimensional sequences is studied. The study establishes results regarding the Poissonian pair correlation of strictly increasing sequences of natural numbers in both supnorm and 2-norm. These results provide a negative answer to a question raised by Hofer and Kaltenbock [15].