Correlations of the Fractional Parts of $\alpha n^\theta$

TL;DR

This paper proves that the fractional parts of lpha n^ for lpha>0 exhibit Poissonian m-point correlations under certain conditions, using novel combinatorial and Fourier analytic methods, advancing understanding of their statistical distribution.

Contribution

It introduces new combinatorial and Fourier analytic techniques to establish Poissonian m-point correlations for fractional parts of lpha n^ sequences, extending previous methods.

Findings

Proves Poissonian m-point correlation for lpha n^ sequences under lpha>0 and <_m.

Develops a new combinatorial argument for higher correlation levels.

Introduces a Fourier analytic approach with an extra frequency variable to analyze oscillatory integrals.

Abstract

Let $m\geq 3$, we prove that $(\alpha n^\theta \mod 1)_{n>0}$ has Poissonian $m$-point correlation for all $\alpha>0$, provided $\theta<\theta_m$, where $\theta_m$ is an explicit bound which goes to $0$ as $m$ increases. This work builds on the method developed in Lutsko-Sourmelidis-Technau (2021), and introduces a new combinatorial argument for higher correlation levels, and new Fourier analytic techniques. A key point is to introduce an `extra' frequency variable to de-correlate the sequence variables and to eventually exploit a repulsion principle for oscillatory integrals. Presently, this is the only positive result showing that the $m$-point correlation is Poissonian for such sequences.