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

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\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 .