paper

Additive energy and the metric Poissonian property

arXiv:1709.02634 · doi:10.1112/S0025579318000207

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.

Slight changes from version 1 based on comments from the referee