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