Note on the Theorem of Balog, Szemerédi, and Gowers

We prove that every additive set A with energy \(E(A)\ge |A|^3/K\) $E(A)\ge {|A|}^3/K$ has a subset \(A'\subseteq A\) $A^{\prime }\subseteq A$ of size \(|A'|\ge (1-\varepsilon )K^{-1/2}|A|\) $|A^{\prime }|\ge (1-\epsilon )K^{-1/2}|A|$ such that \(|A'-A'|\le O_\varepsilon (K^{4}|A'|)\) $|A^{\prime }-A^{\prime }|\le O_{\epsilon }(K^4|A^{\prime }|)$ . This is, essentially, the largest structured set one can get in the Balog–Szemerédi–Gowers theorem.