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Pair correlation of sequences $(\lbrace a_n \alpha \rbrace)_{n \in {\mathbb N}}$ with maximal additive energy

Published online by Cambridge University Press:  06 September 2018

GERHARD LARCHER  and
WOLFGANG STOCKINGER
GERHARD LARCHER
Affiliation:
Department of Financial Mathematics and Applied Number Theory, Johannes Kepler University, Linz, Altenbergerstr. 69, A-4040, Austria. e-mail: [email protected], [email protected]
WOLFGANG STOCKINGER
Affiliation:
Department of Financial Mathematics and Applied Number Theory, Johannes Kepler University, Linz, Altenbergerstr. 69, A-4040, Austria. e-mail: [email protected], [email protected]
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Abstract

We show for sequences $\left(a_{n}\right)_{n \in \mathbb N}$ of distinct positive integers with maximal order of additive energy, that the sequence $\left(\left\{a_{n} \alpha\right\}\right)_{n \in \mathbb N}$ does not have Poissonian pair correlations for any α. This result essentially sharpens a result obtained by J. Bourgain on this topic.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 2 , March 2020 , pp. 287 - 293
Copyright
Copyright © Cambridge Philosophical Society 2018 

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