#### Vol. 5, No. 3, 2012

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On the Bogolyubov–Ruzsa lemma

### Tom Sanders

Vol. 5 (2012), No. 3, 627–655
##### Abstract

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

##### Keywords
Freiman, Fourier analysis, sumsets, generalised arithmetic progressions, coset progressions, small doubling
Primary: 11L07
##### Milestones
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