The number variance of dilated integer sequences

Abstract: Let $(x_n)$ be a sequence of integers. We study the distribution modulo 1 of the dilated sequence $(\alpha x_n)$ in short intervals of length $S$, with the aim of comparing its number variance to that of the random (Poisson) model, thereby identifying regimes of pseudorandom behaviour. We focus on the case of integer polynomial sequences $x_n = p(n)$ of degree at least 2. In joint work with C. Aistleitner (TU Graz), we show that these sequences exhibit Poissonian number variance for almost all $\alpha$. Moreover, this result holds uniformly over a large, and presumably optimal, range of the interval length $S$.

number theory

Audience: researchers in the topic

London number theory seminar

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