On the Bogolyubov–Ruzsa lemma

Abstract

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