Note on the Theorem of Balog, Szemerédi, and Gowers

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