On the correlations of $n^\alpha$ mod 1

TL;DR

This paper investigates the local statistical properties of fractional parts of sequences of the form $n^eta$, establishing that for almost every exponent $eta$ above a certain threshold, the sequence exhibits Poissonian $k$-level correlations, indicating randomness in their distribution.

Contribution

The paper proves that for almost every $eta$ exceeding a specific bound, the $k$-level correlation functions of $ig(ig rbracket n^eta ig rbracketig)$ are Poissonian, advancing understanding of their distributional properties.

Findings

$k$-level correlations are Poissonian for almost every $eta > 4k^2 - 4k - 1$

Sequence fractional parts exhibit randomness in local statistics for large $eta$

Results extend knowledge of uniform distribution to finer statistical levels

Abstract

A well known result in the theory of uniform distribution modulo one (which goes back to Fej\'er and Csillag) states that the fractional parts $\{n^\alpha\}$ of the sequence $(n^\alpha)_{n\ge1}$ are uniformly distributed in the unit interval whenever $\alpha>0$ is not an integer. For sharpening this knowledge to local statistics, the $k$-level correlation functions of the sequence $(\{n^\alpha\})_{n\geq1}$ are of fundamental importance. We prove that for each $k\ge2,$ the $k$-level correlation function $R_k$ is Poissonian for almost every $\alpha>4k^2-4k-1$.