Metric Poissonian pair correlationa and additive energy

TL;DR

This paper proves that sequences with sufficiently low additive energy exhibit Poissonian pair correlation for almost all real numbers, extending previous bounds and providing new insights into the distribution properties of such sequences.

Contribution

It establishes a new lower bound on the additive energy exponent ensuring Poissonian pair correlation for almost all real numbers.

Findings

Sequences with additive energy below N^3/(log N)^C have Poissonian pair correlation.

Provides a lower bound for the exponent C in additive energy bounds.

Extends previous results by Bloom and Walker on pair correlation properties.

Abstract

In this article we prove that if the additive energy of a strictly increasing sequence $(a_n)$ of natural numbers is less than $N^3/(\log N)^C$ for some $C\geq13.155$, then $(\{a_n\alpha\})$ has Poissonian pair correlation for almost all $\alpha\in\mathbb{R}.$ This provides a lower bound for the exponent $C$ in the additive energy bound established by Bloom and Walker[3].