A bilinear Bogolyubov-Ruzsa lemma with poly-logarithmic bounds.

Bogolyubov-Ruzsa lemma, in particular the quantitative bound obtained by Sanders, plays a central role in obtaining effective bounds for the U-3 inverse theorem for the Gowers norms. Recently, Gowers and Milicevic applied a bilinear Bogolyubov-Ruzsa lemma as part of a proof of the U-4 inverse theorem with effective bounds. The goal of this note is to obtain a quantitative bound for the bilinear Bogolyubov-Ruzsa lemma which is similar to that obtained by Sanders for the Bogolyubov-Ruzsa lemma. We show that if a set A subset of F-n x F-n has density a, then after a constant number of horizontal and vertical sums, the set A contains a bilinear structure of codimension r = log(O(1))alpha(-1). This improves the result of Gowers and Milicevic, who obtained a similar statement with a weaker bound of r = exp(exp(log(O(1))alpha(-1))), and by Bienvenu and Le, who obtained r = exp (exp (exp (log(O(1))alpha(-1)))).