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
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.
Original languageEnglish
Pages (from-to)679-700
Number of pages22
JournalMathematika
Volume64
Issue number3
Early online date19 Jun 2018
DOIs
Publication statusPublished - 2018

Bibliographical note

© 2018 University College London. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

Keywords

  • pair correlations
  • distribution modulo 1
  • metric diophantine approximation
  • additive combinatorics
  • large deviations

Cite this