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Remarks about inhomogeneous pair correlations

Part of: Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  06 September 2021

FELIPE A. RAMÍREZ
FELIPE A. RAMÍREZ*
Affiliation:
Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut, U.S.A e-mail: framirez@wesleyan.edu
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Abstract

Given an infinite subset $\mathcal{A} \subseteq\mathbb{N}$, let A denote its smallest N elements. There is a rich and growing literature on the question of whether for typical $\alpha\in[0,1]$, the pair correlations of the set $\alpha A (\textrm{mod}\ 1)\subset [0,1]$ are asymptotically Poissonian as N increases. We define an inhomogeneous generalisation of the concept of pair correlation, and we consider the corresponding doubly metric question. Many of the results from the usual setting carry over to this new setting. Moreover, the double metricity allows us to establish some new results whose singly metric analogues are missing from the literature.

MSC classification

Primary: 11J71: Distribution modulo one 11K38: Irregularities of distribution, discrepancy 11J83: Metric theory
Secondary: 11K60: Diophantine approximation 11K31: Special sequences

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 2 , September 2022 , pp. 369 - 386
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

For Jorge A. Ramírez (1954–2020)—with Love and Gratitude

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