On the number variance of sequences with small additive energy

TL;DR

This paper investigates the variance in the count of fractional parts of sequences with small additive energy, showing Poissonian behavior for almost all scaling factors under certain conditions.

Contribution

It establishes conditions on the additive energy of sequences that ensure Poissonian number variance for almost all real multipliers.

Findings

Number variance asymptotic to L for sequences with bounded additive energy.

Poissonian number variance holds for polynomial sequences with degree ≥ 2.

Results apply when L scales as N^β with β<1/2.

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 $0\le\beta<1/2$.