Effective pair correlations of fractional powers of integers

Abstract

We study the statistics of pairs from the sequence $(n^{\alpha }{)}_{n\in {\mathbb{N}}^{\ast }}$, for every parameter $\alpha \in ]0,1[$. We prove the convergence of the empirical pair correlation measures towards a measure with an explicit density. In particular, when using the scaling factor $N\mapsto N^{1-\alpha }$, 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.

23 pages, 2 figures