The metric theory of the pair correlation function for small non-integer powers

TL;DR

This paper proves that for almost all real numbers, the pair correlation of fractional parts of sequences involving non-integer powers is Poissonian, extending the understanding of distribution properties of such sequences.

Contribution

It establishes the Poissonian nature of pair correlation functions for sequences with fractional powers, a novel result in metric number theory.

Findings

Pair correlation function is Poissonian for almost all α

Results hold for sequences with non-integer powers 0<θ<1

Advances understanding of distribution of fractional parts

Abstract

For $0<\theta<1$, we show that for almost all $\alpha$, the pair correlation function of the sequence of fractional parts of $\{\alpha n^\theta:n\geq 1 \}$ is Poissonian.