On the Bogolyubov–Ruzsa lemma

Sanders, Tom

doi:10.2140/apde.2012.5.627

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On the Bogolyubov–Ruzsa lemma

Tom Sanders

Vol. 5 (2012), No. 3, 627–655

DOI: 10.2140/apde.2012.5.627

Abstract

Our main result is that if $A$ is a finite subset of an abelian group with $| A + A | \leq K | A |$ , then $2 A - 2 A$ contains an $O ({log}^{O (1)} 2 K)$ -dimensional coset progression $M$ of size at least $exp (- O ({log}^{O (1)} 2 K)) | A |$ .

Keywords

Freiman, Fourier analysis, sumsets, generalised arithmetic progressions, coset progressions, small doubling

Mathematical Subject Classification 2010

Primary: 11L07

Milestones

Received: 4 November 2010

Revised: 12 September 2011

Accepted: 9 October 2011

Published: 15 October 2012

Authors

Tom Sanders
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom

Vol. 5, No. 3, 2012