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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
Affiliation:
Department of Financial Mathematics and Applied Number Theory, Johannes Kepler University, Linz, Altenbergerstr. 69, A-4040, Austria. e-mail: wolfgang.stockinger@jku.at, gerhard.larcher@jku.at
WOLFGANG STOCKINGER
Affiliation:
Department of Financial Mathematics and Applied Number Theory, Johannes Kepler University, Linz, Altenbergerstr. 69, A-4040, Austria. e-mail: wolfgang.stockinger@jku.at, gerhard.larcher@jku.at

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
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

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