A bilinear Bogolyubov-Ruzsa lemma with poly-logarithmic bounds

Abstract

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 note 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 $A\subset {\mathbb{F}}_p^n\times {\mathbb{F}}_p^n$ 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 $r=\exp (\exp (\exp ({\log }^{O(1)}{\alpha }^{-1})))$.

14 pages