Additive energy and the metric Poissonian property

TL;DR

This paper investigates the relationship between additive energy of sets of natural numbers and their metric Poissonian property, focusing on how low additive energy influences the property’s occurrence.

Contribution

It advances understanding of how the convergence of additive energy sums determines the metric Poissonian property of sets.

Findings

Explores the convergence aspect of additive energy sums and their impact on metric Poissonian behavior.

Provides theoretical insights into the conditions under which sets exhibit the metric Poissonian property.

Abstract

Let $A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of $A$ (the number of solutions to $a_1 + a_2 = a_3 + a_4$ with $a_i \in A$) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of $A$ modulo $1$. There appears to be reasonable evidence to speculate a sharp Khintchine-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.