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Article contents
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
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
- 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
Footnotes
For Jorge A. Ramírez (1954–2020)—with Love and Gratitude
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