On a theorem of Shkredov

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 $|A cap ext{ Span}(mathcal L)| = Omega (c^{1/3}|A|)$.