On the number variance of sequences with small additive energy

Abstract

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