The Bogolyubov–Ruzsa lemma, in particular the quantitative bounds obtained by Sanders, plays a central role in obtaining effective bounds for the inverse \(U^3\) theorem for the Gowers norms. Recently, Gowers and Milićević applied a bilinear Bogolyubov–Ruzsa lemma as part of a proof of the inverse \(U^4\) theorem with effective bounds. The goal of this talk is to obtain quantitative bounds for the bilinear Bogolyubov–Ruzsa lemma which are similar to those obtained by Sanders for the Bogolyubov–Ruzsa lemma. We show that if a set \(A \subset \mathbb{F}^n \times \mathbb{F}^n\) has density \(\alpha\), then after a constant number of horizontal and vertical sums, the set \(A\) would contain a bilinear structure of co-dimension \(r=\log^{O(1)} \alpha^{-1}\). This improves the results of Gowers and Milićević which obtained similar results with a weaker bound of \(r=\exp(\exp(\log^{O(1)} \alpha^{-1}))\) and by Bienvenu and Lê which obtained \(r=\exp(\exp(\exp(\log^{O(1)} \alpha^{-1})))\).
Joint work with Shachar Lovett.
Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.
The ACO Seminar (2019–2020)