# Linear forms and quadratic uniformity for functions on ℤ_{ N }

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## 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.

## Keywords

Quadratic Form Linear Form Bilinear Form Phase Function Triangle Inequality## Preview

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## References

- [B99]J. Bourgain,
*On triples in arithmetic progression*, Geom. Funct. Anal.**9**(1999), 968–984.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar - [F73]G. Freiman,
*Foundations of a Structural Theory of Set Addition*, Amer. Math. Soc., Providence, RI, 1973.zbMATHGoogle Scholar - [G01]W. T. Gowers,
*A new proof of Szemerédi’s theorem*, Geom. Funct. Anal.**11**(2001), 465–588.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar - [GW10b]W. T. Gowers and J. Wolf,
*Linear forms and quadratic uniformity for functions on*F_{p}^{n}, Mathematika**57**(2012), 215–237.CrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar - [GrT10a]B. J. Green and T. Tao,
*Linear equations in primes*, Annals of Math. (2)**171**(2010), 1753–1850.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar - [TV06]T. Tao and V. Vu.
*Additive Combinatorics*, Cambridge University Press, Cambridge, 2006.zbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar