Linear forms and quadratic uniformity for functions on ℤ_{N} Article First Online: 16 November 2011 Received: 06 May 2010 Revised: 17 November 2010 DOI :
10.1007/s11854-011-0026-7

Cite this article as: Gowers, W.T. & Wolf, J. JAMA (2011) 115: 121. doi:10.1007/s11854-011-0026-7
4
Citations
143
Downloads
Abstract A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW10a] and [GW10b] , [GW10c] , we gave a complete answer to this question for groups of the form G = F _{p} ^{n} , provided p is not too small. In ℤ_{N} , substantial extra difficulties arise, of which the most important is that an “inverse theorem” even for the uniformity norm \({\left\| \cdot \right\|_{{U^3}}}\) requires a more sophisticated “local” formulation. When N is prime, ℤ_{N} is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW10a] . Moreover, we obtain a doubly exponential bound.

References [B99]

J. Bourgain,

On triples in arithmetic progression , Geom. Funct. Anal.

9 (1999), 968–984.

MathSciNet MATH CrossRef Google Scholar [BTZ10]

V. Bergelson, T. Tao, and T. Ziegler.

An inverse theorem for the uniformity seminorms associated with the action of
F
_{p} ^{ω} , Geom. Funct. Anal.

19 (2010), 1539–1596.

MathSciNet MATH CrossRef Google Scholar [C10]

P. Candela,

On the structure of steps of three-term arithmetic progressions in a dense set of integers , Bull. London Math. Soc.

42 (2010), 1–14.

MathSciNet MATH CrossRef Google Scholar [F73]

G. Freiman,

Foundations of a Structural Theory of Set Addition , Amer. Math. Soc., Providence, RI, 1973.

MATH Google Scholar [G01]

W. T. Gowers,

A new proof of Szemerédi’s theorem , Geom. Funct. Anal.

11 (2001), 465–588.

MathSciNet MATH CrossRef Google Scholar [GW10a]

W. T. Gowers and J. Wolf,

The true complexity of a system of linear equations . Proc. London Math. Soc.

100 (2010), 155–176.

MathSciNet MATH CrossRef Google Scholar [GW10b]

W. T. Gowers and J. Wolf,

Linear forms and quadratic uniformity for functions on
F
_{p} ^{n} , Mathematika

57 (2012), 215–237.

CrossRef Google Scholar [GW10c]

W. T. Gowers and J. Wolf,

Linear forms and higher-degree uniformity for functions on
F
_{p} ^{n} , Geom. Funct. Anal.

21 (2011), 36–69.

MathSciNet MATH CrossRef Google Scholar [Gr07]

B. J. Green,

Montreal notes on quadratic Fourier analysis in

Additive Combinatorics , Amer. Math. Soc., Providence, RI, 2007, pp. 69–102.

Google Scholar [GrS07]

B. J. Green and T. Sanders,

A quantitative version of the idempotent theorem in harmonic analysis , Ann. of Math. (2)

168 (2007), 1025–1054.

MathSciNet CrossRef Google Scholar [GrT08]

B. J. Green and T. Tao,

An inverse theorem for the Gowers U
^{3} (

G )

norm , Proc. Edinburgh Math. Soc.

51 (2008), 73–153.

MathSciNet MATH CrossRef Google Scholar [GrT09]

B. J. Green and T. Tao,

New bounds for Szemerédi’s theorem, I: Progressions of length 4 in finite field geometries , Proc. London Math. Soc. (3)

98 (2009), 365–392.

MathSciNet MATH CrossRef Google Scholar [GrT10a]

B. J. Green and T. Tao,

Linear equations in primes , Annals of Math. (2)

171 (2010), 1753–1850.

MathSciNet MATH CrossRef Google Scholar [GrT10b]

B. J. Green and T. Tao,

An arithmetic regularity lemma, an associated counting lemma, and applications , in

An Irregular Mind (Szemerédi is 70) , Springer, 2010, pp. 261–334.

Google Scholar [GrTZ10]

B. J. Green, T. Tao and T. Ziegler,

An inverse theorem for the Gowers U
^{s+1} [

N ]

norm , arXiv:1009.3998v3.

Google Scholar [L10]

A. Leibman,

Orbit of the diagonal of a power of a nilmanifold , Trans. Amer. Math. Soc.

362 (2010), 1619–1658.

MathSciNet MATH CrossRef Google Scholar [N96]

M. B. Nathanson.

Additive Number Theory: Inverse Problems and the Geometry of Sumsets , Springer, New York, 1996.

Google Scholar [R94]

I. Z. Ruzsa.

Generalized arithmetical progressions and sumsets , Acta Math. Hungar.

65 (1994), 379–388.

MathSciNet MATH CrossRef Google Scholar [TV06]

T. Tao and V. Vu.

Additive Combinatorics , Cambridge University Press, Cambridge, 2006.

MATH CrossRef Google Scholar [TZ10]

T. Tao and T. Ziegler,

The inverse conjecture for the Gowers norm over finite fields via the correspondence principle , Anal. PDE

3 (2010), 1–20.

MathSciNet MATH CrossRef Google Scholar © Hebrew University Magnes Press 2011

Authors and Affiliations 1. Department of Pure mathematics and Mathematical Statistics Cambridge UK 2. Department of Mathematics Rutgers The State University of New Jersey Piscataway USA