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

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 $\alpha$. This result essentially sharpens a result obtained by J. Bourgain on this topic.

9 pages. arXiv admin note: text overlap with arXiv:1711.08663, We want to thank an anonymous referee for pointing out an inaccuracy in the first version of the paper