Metric Poissonian pair correlationa and additive energy

Abstract

In this article we prove that for a strictly increasing sequence $(a_n)$ of natural numbers, if the additive energy of $\{a_n:n\le N\}$ is less than $N^3/(\log N{)}^C$ for some $C\ge 14.71,$ then $(\{a_n\alpha \})$ has Poissonian pair correlation for almost all $\alpha \in \mathbb{R}.$ This provides a lower bound for the exponent $C$ in the additive energy bound established by Bloom and Walker [4].

arXiv commentCorrected the lower bound of C and some typos. To appear in Archiv der Mathematik