TR18-142 Authors: Kaave Hosseini, Shachar Lovett

Publication: 17th August 2018 05:30

Downloads: 518

Keywords:

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 Mili\'cevi\'c

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 $A \subset \mathbb{F}^n \times \mathbb{F}^n$

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 Mili\'cevi\'c which obtained similar results with a weaker bound of

$r=\exp(\exp(\log^{O(1)} \alpha^{-1}))$ and by Bienvenu and L\^e which obtained $r=\exp(\exp(\exp(\log^{O(1)} \alpha^{-1})))$.