  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 $\alpha>4k^2-4k-1$.
