Effective pair correlations of fractional powers of integers

TL;DR

This paper investigates the statistical behavior of pairs from fractional power sequences, proving convergence to explicit measures and revealing phenomena like level repulsion under specific scalings.

Contribution

It establishes the convergence of pair correlation measures for fractional power sequences and identifies conditions leading to Poissonian or degenerate correlations.

Findings

Existence of an explicit density for the pair correlation measure.

Identification of a level repulsion phenomenon at specific scalings.

Conditions under which pair correlations are Poissonian or degenerate.

Abstract

We study the statistics of pairs from the sequence $(n^\alpha)_{n\in\mathbb{N}^*}$, 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.