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

Abstract

A well known result in the theory of uniform distribution modulo 1 (which goes back to Fejér and Csillag) states that the fractional parts $\{n^{\alpha }\}$ of the sequence $(n^{\alpha }{)}_{n\ge 1}$ 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 $(\{n^{\alpha }\}{)}_{n\ge 1}$ are of fundamental importance. We prove that for each $k\ge 2,$ the $k$-level correlation function $R_k$ is Poissonian for almost every $\alpha >4k^2-4k-1$.