Linear forms and quadratic uniformity for functions on Fₚⁿ

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We give improved bounds for our theorem in [GW09], which shows that a system of linear forms on Fₚⁿ with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of Fₚⁿ. While in [GW09] 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 [GrT08], we use the Hahn-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 [GrT08].