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ADDITIVE ENERGY AND THE METRIC POISSONIAN PROPERTY

Part of: Diophantine approximation, transcendental number theory Sequences and sets Limit theorems Designs and configurations

Published online by Cambridge University Press:  19 June 2018

Thomas F. Bloom ,
Sam Chow ,
Ayla Gafni  and
Aled Walker
Thomas F. Bloom
Affiliation:
Heilbronn Institute for Mathematical Research, Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, U.K. email matfb@bristol.ac.uk
Sam Chow
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, U.K. email sam.chow@york.ac.uk
Ayla Gafni
Affiliation:
Department of Mathematics, 915 Hylan Building, University of Rochester, Rochester, NY 14627, U.S.A. email agafni@ur.rochester.edu
Aled Walker
Affiliation:
Andrew Wiles Building, University of Oxford, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6GG, U.K. email walker@maths.ox.ac.uk
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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.

MSC classification

Primary: 11J71: Distribution modulo one 11J83: Metric theory 05B10: Difference sets (number-theoretic, group-theoretic, etc.) 11B30: Arithmetic combinatorics; higher degree uniformity 60F10: Large deviations
Type
Research Article
Information
Mathematika , Volume 64 , Issue 3 , 2018 , pp. 679 - 700
Copyright
Copyright © University College London 2018 

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