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

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