A bilinear BogolyubovRuzsa lemma with polylogarithmic bounds
Abstract
The BogolyubovRuzsa 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 BogolyubovRuzsa 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 BogolyubovRuzsa lemma which are similar to those obtained by Sanders for the BogolyubovRuzsa 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 codimension $r=\log^{O(1)} \alpha^{1}$. 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 $r=\exp(\exp(\exp(\log^{O(1)} \alpha^{1})))$.
 Publication:

arXiv eprints
 Pub Date:
 August 2018
 DOI:
 10.48550/arXiv.1808.04965
 arXiv:
 arXiv:1808.04965
 Bibcode:
 2018arXiv180804965H
 Keywords:

 Mathematics  Combinatorics;
 05D99
 EPrint:
 14 pages