On higher dimensional Poissonian pair correlation

In this article we study the pair correlation statistic for higher dimensional sequences. We show that for any \(d\geq 2\), strictly increasing sequences \((a_n^{(1)}),\ldots, (a_n^{(d)})\) of natural numbers have metric Poissonian pair correlation with respect to sup-norm if their joint additive en...

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Bibliographic Details
Published in:arXiv.org 2023-08
Main Authors: Bera, Tanmoy, Das, Mithun Kumar, Mukhopadhyay, Anirban
Format: Article
Language:English
Subjects:
Number theory
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Summary:In this article we study the pair correlation statistic for higher dimensional sequences. We show that for any \(d\geq 2\), strictly increasing sequences \((a_n^{(1)}),\ldots, (a_n^{(d)})\) 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 \((\{n\alpha\}, \{[n\log^An]\beta\})\) (\(A \in [1,2]\)) 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 Kaltenb\"ock [15]. The proof uses estimates for 'Generalized' GCD-sums.
ISSN:2331-8422