We give improved bounds for our theorem in [W.Â T.Â Gowers and J.Â Wolf, The true complexity of a system of linear equations. Proc.Â LondonÂ Math. Soc.Â (3) 100 (2010), 155â€“176], which shows that a system of linear forms on đť”˝np with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of đť”˝np. While in [W.Â T.Â Gowers and J.Â Wolf, The true complexity of a system of linear equations. Proc.Â LondonÂ Math.Â Soc.Â (3) 100 (2010), 155â€“176] the dependence between the uniformity of the set and the resulting error in the average over the linear system was of tower type, we now obtain a doubly exponential relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [An inverse theorem for the Gowers U3(G) norm. Proc.Â Edinb.Â Math.Â Soc.Â (2) 51 (2008), 73â€“153], we use the Hahnâ€“Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the U3 inverse theorem [B.Â J.Â Green and T.Â Tao, An inverse theorem for the Gowers U3(G) norm. Proc.Â Edinb.Â Math.Â Soc.Â (2) 51 (2008), 73â€“153].
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