Small-difference-set form of the Balog-Szemerédi-Gowers theorem
= Small-difference-set form of the Balog-Szemerédi-Gowers theorem
If $E(A)\geq\theta|A|^3$, a subset $B\subseteq A$ satisfies $|B|\geq c_\theta|A|$ and $|B-B|\leq K_\theta|B|$. A <dependent random choice> argument in the <popular sum> graph finds many four-edge paths representing each member of the <difference set>. The <Petridis minimal-growth lemma> then bounds all higher <iterated sumsets> and <difference sets>.