Pair correlation of sequences $(\lbrace a_n \alpha \rbrace)_{n \in \mathbb{N}}$ 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.