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 F n F n has density , 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) ? 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) ? 1 )) and by Bienvenu and L^e which obtained r = exp ( exp ( exp ( log O (1) ? 1 ))) .
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