Article contents
ADDITIVE ENERGY AND THE METRIC POISSONIAN PROPERTY
Published online by Cambridge University Press: 19 June 2018
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 Khinchin-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.
- Type
- Research Article
- Information
- Copyright
- Copyright © University College London 2018
References
- 11
- Cited by