On the correlations of $n^α$ mod 1

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

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