Note on the Theorem of Balog, Szemerédi, and Gowers

Abstract

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

second version addresses referee reports

References in corpus (1)

• Large values of the additive energy in R^d and Z^d