On a theorem of Shkredov

Abstract

We show that if $A$ is a finite subset of an abelian group with additive energy at least $c|A{|}^3$, then there is a set $\mathcal{L}\subset A$ with $|\mathcal{L}|=O(c^{-1}\log |A|)$ such that $|A\cap \mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}(\mathcal{L})|=\mathrm{\Omega }(c^{1/3}|A|)$.