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  • W. T. Gowers (a1) and J. Wolf (a2)

We give improved bounds for our theorem in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176], which shows that a system of linear forms on 𝔽np with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of 𝔽np. While in [W. T. Gowers and J. Wolf, The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155–176] 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. Edinb. Math. Soc. (2) 51 (2008), 73–153], 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 [B. J. Green and T. Tao, An inverse theorem for the Gowers U3(G) norm. Proc. Edinb. Math. Soc. (2) 51 (2008), 73–153].

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[1]Bourgain, J., On triples in arithmetic progression. Geom. Funct. Anal. 9(5) (1999), 968984.
[2]Candela, P., A U 3 inverse theorem for the P U 3 norm. Preprint, 2007.
[3]Gowers, W. T., Decompositions, approximate structure, transference, and the Hahn–Banach theorem. Bull. London Math. Soc. 42(4) (2010), 573606.
[4]Gowers, W. T. and Wolf, J., Linear forms and higher-degree uniformity for functions on 𝔽np. Geom. Funct. Anal. (to appear).
[5]Gowers, W. T. and Wolf, J., Linear forms and quadratic uniformity for functions on ℤN. J. Anal. Math. (to appear).
[6]Gowers, W. T. and Wolf, J., The true complexity of a system of linear equations. Proc. London Math. Soc. (3) 100 (2010), 155176.
[7]Green, B. J., Finite field models in additive combinatorics. In Surveys in Combinatorics 2005 (London Mathematical Society Lecture Notes 327), Cambridge University Press (2005), 1–27.
[8]Green, B. J. and Tao, T., An inverse theorem for the Gowers U 3(G) norm. Proc. Edinb. Math. Soc. (2) 51 (2008), 73153.
[9]Green, B. J. and Tao, T., The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) 167 (2008), 481547.
[10]Green, B. J. and Tao, T., New bounds for Szemerédi’s theorem, I: progressions of length 4 in finite field geometries. Proc. London Math. Soc. (3) 98 (2009), 365392.
[11]Green, B. J. and Tao, T., An equivalence between inverse sumset theorems and inverse conjectures for the U 3 norm. Math. Proc. Cambridge Philos. Soc. 149(1) (2010), 119.
[12]Green, B. J. and Tao, T., An arithmetic regularity lemma, an associated counting lemma, and applications. In An Irregular Mind: Szemeredi is 70 (Bolyai Society Mathematical Studies 21), Springer (2010).
[13]Green, B. J. and Tao, T., Linear equations in primes. Ann. of Math. (2) 171 (2010), 17531850.
[14]Samorodnitsky, A., Low-degree tests at large distances. Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, ACM (New York, NY, 2007), 506–515.
[15]Tao, T. and Vu, V., Additive Combinatorics, Cambridge University Press (Cambridge, 2006).
[16]Wolf, J., A local inverse theorem in 𝔽n2. Preprint, 2009.
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  • ISSN: 0025-5793
  • EISSN: 2041-7942
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