# Linear Forms and Higher-Degree Uniformity for Functions On \({\mathbb{F}^{n}_{p}}\)

- 151 Downloads
- 12 Citations

## Abstract

In [GW1] we began an investigation of the following general question. Let *L* _{1}, . . . , *L* _{ m } be a system of linear forms in *d* variables on \({F^n_p}\), and let *A* be a subset of \({F^n_p}\) of positive density. Under what circumstances can one prove that *A* contains roughly the same number of *m*-tuples *L* _{1}(*x* _{1}, . . . , *x* _{ d }), . . . , *L* _{ m }(*x* _{1}, . . . , *x* _{ d }) with \({x_1,\ldots, x_d \in {\mathbb F}^n_p}\) as a typical random set of the same density? Experience with arithmetic progressions suggests that an appropriate assumption is that \({||A - \delta 1||_{U{^k}}}\) should be small, where we have written *A* for the characteristic function of the set *A*, *δ* is the density of *A*, *k* is some parameter that depends on the linear forms *L* _{1}, . . . , *L* _{ m }, and \({|| \cdot ||_U{^k}}\) is the *k*th uniformity norm. The question we investigated was how *k* depends on *L* _{1}, . . . , *L* _{ m }. Our main result was that there were systems of forms where *k* could be taken to be 2 even though there was no simple proof of this fact using the Cauchy–Schwarz inequality. Based on this result and its proof, we conjectured that uniformity of degree *k* − 1 is a sufficient condition if and only if the *k*th powers of the linear forms are linearly independent. In this paper we prove this conjecture, provided only that *p* is sufficiently large. (It is easy to see that some such restriction is needed.) This result represents one of the first applications of the recent inverse theorem for the *U* ^{ k } norm over \({F^n_p}\) by Bergelson, Tao and Ziegler [TZ2], [BTZ]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.

## Keywords and phrases

Higher order Fourier analysis uniformity norms solutions to systems of linear equations## 2010 Mathematics Subject Classification

11B30## Preview

Unable to display preview. Download preview PDF.

## References

- BTZ.Bergelson V., Tao T., Ziegler T.: An inverse theorem for the uniformity seminorms associated with the action of \({F^\omega_p}\). Geom. Funct. Anal.
**19**(6), 1539–1596 (2010)MathSciNetzbMATHCrossRefGoogle Scholar - G1.Gowers W.T.: A new proof of Szemerédi’s theorem. Geom. Funct. Anal.
**11**, 465–588 (2001)MathSciNetzbMATHCrossRefGoogle Scholar - G2.Gowers W.T.: Decompositions, approximate structure, transference, and the Hahn–Banach theorem. Bull. London Math. Soc.
**42**(4), 573–606 (2010)MathSciNetzbMATHCrossRefGoogle Scholar - GW1.Gowers W.T., Wolf J.: The true complexity of a system of linear equations. Proc. London Math. Soc.
**100**(3), 155–176 (2010)MathSciNetzbMATHCrossRefGoogle Scholar - GW2.W.T. Gowers, J. Wolf, Linear forms and quadratic uniformity for functions on \({\mathbb{F}^{n}_{p}}\), Arxiv preprint arXiv:1002.2209 (2010).Google Scholar
- GW3.W.T. Gowers, J. Wolf, Linear forms and quadratic uniformity for functions on ZN, Arxiv preprint arXiv:1002.2210 (2010).Google Scholar
- Gr1.Green B.J.: A Szemerédi-type regularity lemma in abelian groups. Geom. Funct. Anal.
**15**, 340–376 (2005)MathSciNetzbMATHCrossRefGoogle Scholar - Gr2.B.J. Green, Montreal lecture notes on quadratic Fourier analysis. in “Additive Combinatorics (Montréal 2006, (Granville, et al., eds), CRM Proceedings 43, AMS (2007), 69–102.Google Scholar
- GrT1.Green B.J., Tao T.: An inverse theorem for the Gowers
*U*^{3}(*G*) norm. Proc. Edinburgh Math. Soc.**51**, 73–153 (2008)MathSciNetzbMATHCrossRefGoogle Scholar - GrT2.Green B.J., Tao T.: The primes contain arbitrarily long arithmetic progressions. Annals of Math.
**167**, 481–547 (2008)MathSciNetzbMATHCrossRefGoogle Scholar - GrT3.Green B.J., Tao T.: The distribution of polynomials over finite fields. with applications to the Gowers norms, Contrib. Discrete Math
**4**(2), 1–36 (2009)MathSciNetzbMATHGoogle Scholar - GrT4.Green B.J., Tao T.: Linear equations in primes. Annals of Math.
**171**, 1753–1850 (2010)MathSciNetzbMATHCrossRefGoogle Scholar - GrT5.B.J. Green, T. Tao, An arithmetic regularity lemma, an associated counting lemma, and applications, in “An Irregular Mind: Szemeredi is 70”, Bolyai Society Math. Studies 21 (2010).Google Scholar
- GrTZ.B.J. Green, T. Tao, T. Ziegler, An inverse theorem for the Gowers
*U*^{s+1}[*N*] norm, Arxiv preprint arXiv:1009.3998 (2010).Google Scholar - L.Leibman A.: Orbit of the diagonal of a power of a nilmanifold. Trans. Amer. Math. Soc.
**362**, 1619–1658 (2010)MathSciNetzbMATHCrossRefGoogle Scholar - LoMS.S. Lovett, R. Meshulam, A. Samorodnitsky, Inverse Conjecture for the Gowers norm is false, in “Proceedings of the 40th Annual ACM Symposium on Theory of Computing”, ACM New York, NY, USA (2008), 547–556.Google Scholar
- S.A. Samorodnitsky, Low-degree tests at large distances, in Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, ACM New York, NY, USA (2007), 506–515.Google Scholar
- T.T. Tao, A quantitative ergodic theory proof of Szemerédi’s theorem, Electron. J. Combin. 13:1 (2006), Research Paper 99 (2006).Google Scholar
- TZ1.Tao T., Ziegler T.: The primes contain arbitrarily long polynomial progressions. Acta Mathematica
**201**(2), 213–305 (2008)MathSciNetzbMATHCrossRefGoogle Scholar - TZ2.Tao T., Ziegler T.: The inverse conjecture for the Gowers norm over finite fields via the correspondence principle. Analysis & PDE
**3**, 1–20 (2010)MathSciNetCrossRefGoogle Scholar