Additive energy and the metric Poissonian property

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JournalMathematika
DateAccepted/In press - 11 Feb 2018
DateE-pub ahead of print (current) - 19 Jun 2018
Issue number3
Volume64
Number of pages22
Pages (from-to)679-700
Early online date19/06/18
Original languageEnglish

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.

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© 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

    Research areas

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

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