On the number variance of sequences with small additive energy

For a real-valued sequence $(x_n{)}_{n=1}^{\mathrm{\infty }}$, denote by $S_N(\mathrm{\ell })$ the number of its first $N$ fractional parts lying in a random interval of size $\mathrm{\ell }:=L\mathrm{/}N$, where $L=o(N)$ as $N\to \mathrm{\infty }$. We study the variance of $S_N(\mathrm{\ell })$ (the number variance) for sequences of the form $x_n=\alpha a_n$, where $(a_n{)}_{n=1}^{\mathrm{\infty }}$ is a sequence of distinct integers. We show that if the additive energy of the sequence $(a_n{)}_{n=1}^{\mathrm{\infty }}$ is bounded from above by $N^{5\mathrm{/}2-\epsilon }\mathrm{/}L$ for some $\epsilon >0$, then for almost all $\alpha$, the number variance is asymptotic to $L$ (Poissonian number variance). This holds in particular for the sequence $x_n=\alpha n^d,d\ge 2$ whenever $L=N^{\beta }$ with $0\le \beta <1\mathrm{/}2$.