On an almost all version of the Balog-Szemeredi-Gowers theorem

Abstract

We deduce, as a consequence of the arithmetic removal lemma, an almost-all version of the Balog-Szemer\'{e}di-Gowers theorem: For any $K\geq 1$ and $\varepsilon > 0$, there exists $\delta = \delta(K,\varepsilon)>0$ such that the following statement holds: if $|A+_{\Gamma}A| \leq K|A|$ for some $\Gamma \geq (1-\delta)|A|^2$, then there is a subset $A' \subset A$ with $|A'| \geq (1-\varepsilon)|A|$ such that $|A'+A'| \leq |A+_{\Gamma}A| + \varepsilon |A|$. We also discuss issues around quantitative bounds in this statement, in particular showing that when $A \subset \mathbb{Z}$ the dependence of $\delta$ on $\epsilon$ cannot be polynomial for any fixed $K>2$.