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arXiv · 2504.00708

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

Christoph Aistleitner·Nadav Yesha
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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 $0 \leq S \leq (\log N)^{-c}$, 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 $(x_n)_{n=1}^N$.

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Christoph Aistleitner, Nadav Yesha. 2025-04-01. Almost sure asymptotics for the number variance of dilations of integer sequences. https://arxiv.org/abs/2504.00708

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