On the number variance of sequences with small additive energy

For a real-valued sequence ${\left(x_n\right)}_{n=1}^{\infty }$, denote by $S_N\left(\ell \right)$ the number of its first N fractional parts lying in a random interval of size $\ell :=L/N$, where $L=o\left(N\right)$ as $N\to \infty$. We study the variance of $S_N\left(\ell \right)$ (the number variance) for sequences of the form $x_n=\alpha a_n$, where ${\left(a_n\right)}_{n=1}^{\infty }$ is a sequence of distinct integers. We show that if the additive energy of the sequence ${\left(a_n\right)}_{n=1}^{\infty }$ is bounded from above by $N^{3-\epsilon }/L^2$ for some $\epsilon >0$, then for almost all α, 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 $0\le \beta <1/2$.