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.
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 language | English |
|---|---|
| Pages (from-to) | 679-700 |
| Number of pages | 22 |
| Journal | Mathematika |
| Volume | 64 |
| Issue number | 3 |
| Early online date | 19 Jun 2018 |
| DOIs | |
| Publication status | Published - 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 detailsKeywords
- pair correlations
- distribution modulo 1
- metric diophantine approximation
- additive combinatorics
- large deviations