Almost sure asymptotics for the number variance of dilations of integer sequences

Abstract

Let $(x_n{)}_{n=1}^{\infty }$ be a sequence of integers. We study the number variance of dilations $(\alpha x_n{)}_{n=1}^{\infty }$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $\alpha$ throughout a large range of $S$, subject to certain regularity assumptions imposed upon $(x_n{)}_{n=1}^{\infty }$. For the important special case $x_n=p(n)$, where $p$ is a polynomial with integer coefficients of degree at least 2, we prove that the number variance is Poissonian for almost all $\alpha$ throughout the range $0\le S\le (\log N{)}^{-c}$, for a suitable absolute constant $c>0$. For more general sequences $(x_n{)}_{n=1}^{\infty }$, we give a criterion for Poissonian behavior for generic $\alpha$ which is formulated in terms of the additive energy of the finite truncations $(x_n{)}_{n=1}^N$.

37 pages, 1 figure