Metric Poissonian pair correlationa and additive energy

In this article we prove that if the additive energy of a strictly increasing sequence $(a_n)$ of natural numbers is less than $N^3\mathrm{/}(\log N{)}^C$ for some $C\ge 13.155$, then $(\{a_n\alpha \})$ has Poissonian pair correlation for almost all $\alpha \in \mathbb{R}\mathrm{.}$ This provides a lower bound for the exponent $C$ in the additive energy bound established by Bloom and Walker[3].