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LINEAR FORMS AND QUADRATIC UNIFORMITY FOR FUNCTIONS ON 𝔽np

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

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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[15] T. Tao and V. Vu , Additive Combinatorics, Cambridge University Press (Cambridge, 2006).

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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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