Remark on a result of Bourgain on poissonian pair correlation

Abstract

We show for a class of sequences $(a_n{)}_{n\ge 1}$ of distinct positive integers, that for no $\alpha$ the sequence $(\left\{a_n\alpha \right\}{)}_{n\ge 1}$ does have Poissonian pair correlation. This class contains for example all strictly increasing integer sequences with positive upper density. This result motivates us to state a certain conjecture on Poissonian pair correlation which would be a significantly stronger version of a result of Jean Bourgain.

arXiv comment2 figures, 13 pages

References in corpus (1)

• The distribution of spacings between the fractional parts of $n^2\alpha$

Cited by in corpus (1)

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