Intermediate-scale statistics for real-valued lacunary sequences

TL;DR

This paper investigates the statistical distribution of fractional parts of lacunary sequences, showing they behave similarly to random points in the unit interval at intermediate scales, with proven variance asymptotics and a central limit theorem.

Contribution

It establishes variance asymptotics and a central limit theorem for the fractional parts of lacunary sequences at intermediate scales, extending understanding of their statistical properties.

Findings

Variance of element count matches that of uniform random points.

Asymptotic behavior holds almost surely in the parameter α.

Central limit theorem applies for slowly growing interval lengths.

Abstract

We study intermediate-scale statistics for the fractional parts of the sequence $(\alpha a_n)_{n=1}^{\infty}$, where $(a_n)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$. In particular, we consider the number of elements $S_{N}(L,\alpha)$ in a random interval of length $L/N$, where $L=O\left(N^{1-\epsilon}\right)$, and show that its variance (the number variance) is asymptotic to $L$ with high probability w.r.t. $\alpha$, which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotics holds almost surely in $\alpha\in\mathbb{R}$ when $L=O\left(N^{1/2-\epsilon}\right)$. For slowly growing $L$, we further prove a central limit theorem for $S_{N}(L,\alpha)$ which holds for almost all $\alpha\in\mathbb{R}$.