Linear forms and quadratic uniformity for functions on ℤN

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 𝔽 n p with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of 𝔽 n p . 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 U 3 (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 U 3 inverse theorem [B. J. Green and T. Tao, An inverse theorem for the Gowers U 3 (G) norm. Proc. Edinb. Math. Soc.  (2) 51 (2008), 73–153].

[1]  G. Freiman.  Foundations of a Structural Theory of Set Addition , 2007 .

[2]  T. Sanders,et al.  A quantitative version of the idempotent theorem in harmonic analysis , 2006, math/0611286.

[3]  W. T. Gowers,et al.  Decompositions, approximate structure, transference, and the Hahn–Banach theorem , 2008, 0811.3103.

[4]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS U4-NORM , 2005, Glasgow Mathematical Journal.

[5]  Ben Green,et al.  An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications , 2010, 1002.2028.

[6]  Terence Tao,et al.  Additive combinatorics , 2007, Cambridge studies in advanced mathematics.

[7]  Julia Wolf,et al.  Linear forms and quadratic uniformity for functions on ℤN , 2010, 1002.2210.

[8]  Jean Bourgain,et al.  On Triples in Arithmetic Progression , 1999 .

[9]  Ben Green,et al.  Finite field models in additive combinatories , 2004, BCC.

[10]  Ben Green,et al.  Linear equations in primes , 2006, math/0606088.

[11]  W. T. Gowers,et al.  The true complexity of a system of linear equations , 2007, 0711.0185.

[12]  Ben Green,et al.  New bounds for Szemeredi's theorem, II: A new bound for r_4(N) , 2006 .

[13]  W. T. Gowers,et al.  A new proof of Szemerédi's theorem , 2001 .

[14]  Pablo Candela,et al.  On the structure of steps of three‐term arithmetic progressions in a dense set of integers , 2010 .

[15]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM , 2008, Proceedings of the Edinburgh Mathematical Society.

[16]  Melvyn B. Nathanson,et al.  Additive Number Theory: Inverse Problems and the Geometry of Sumsets , 1996 .

[17]  Ben Green,et al.  An equivalence between inverse sumset theorems and inverse conjectures for the U3 norm , 2009, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  Ben Green,et al.  New bounds for Szemerédi's theorem, I: progressions of length 4 in finite field geometries , 2009 .

[19]  W. T. Gowers,et al.  Linear Forms and Higher-Degree Uniformity for Functions On $${\mathbb{F}^{n}_{p}}$$ , 2010, 1002.2208.

[20]  W. T. Gowers,et al.  A NEW PROOF OF SZEMER ´ EDI'S THEOREM , 2001 .

[21]  Terence Tao,et al.  The inverse conjecture for the Gowers norm over finite fields via the correspondence principle , 2008, 0810.5527.

[22]  A. Leibman.  Orbit of the diagonal in the power of a nilmanifold , 2009 .

[23]  T. Tao,et al.  The primes contain arbitrarily long arithmetic progressions , 2004, math/0404188.

[24]  Ben Green,et al.  Montreal Lecture Notes on Quadratic Fourier Analysis , 2006 .

[25]  Imre Z. Ruzsa,et al.  Generalized arithmetical progressions and sumsets , 1994 .

[26]  Alex Samorodnitsky,et al.  Low-degree tests at large distances , 2006, STOC '07.