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-Szemerédi-Gowers theorem: For any $K\ge 1$ and $\epsilon >0$, there exists $\delta =\delta (K,\epsilon )>0$ such that the following statement holds: if $|A{+}_{\Gamma }A|\le K|A|$ for some $\Gamma \ge (1-\delta )|A{|}^2$, then there is a subset $A^{\prime }\subset A$ with $|A^{\prime }|\ge (1-\epsilon )|A|$ such that $|A^{\prime }+A^{\prime }|\le |A{+}_{\Gamma }A|+\epsilon |A|$. 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 $K>2$.

18 pages

References in corpus (2)

• Progression-free sets in Z_4^n are exponentially small
• A robust version of Freiman's $3k-4$ Theorem and applications