# Additive energy and the metric Poissonian property

Thomas F. Bloom, Samuel Khai Ho Chow, Ayla Gafni, Aled Walker

Research output: Contribution to journalArticlepeer-review

## 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
theory, in other words the extent to which having a low additive energy forces
a set to be metric Poissonian.
Original language English 679-700 22 Mathematika 64 3 19 Jun 2018 https://doi.org/10.1112/S0025579318000207 Published - 2018

### Bibliographical note

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## Keywords

• pair correlations
• distribution modulo 1
• metric diophantine approximation