Effective pair correlations of fractional powers of integers

We study the statistics of pairs from the sequence ( n α ) n ∈ ℕ * , for every parameter α ∈ ] 0 , 1 [ . We prove the convergence of the empirical pair correlation measures towards a measure with an explicit density. In particular, when using the scaling factor N ↦ N 1 - α , we prove that there exists an exotic pair correlation function which exhibits a level repulsion phenomenon. For other scaling factors, we prove that either the pair correlations are Poissonian or there is a total loss of mass. In addition, we give an error term for this convergence.