Note on the Theorem of Balog, Szemerédi, and Gowers

Abstract

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