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

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 kth 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 kth 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.

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References

  1. 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)

    MathSciNet  MATH  Article  Google Scholar 

  2. G1

    Gowers W.T.: A new proof of Szemerédi’s theorem. Geom. Funct. Anal. 11, 465–588 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  3. G2

    Gowers W.T.: Decompositions, approximate structure, transference, and the Hahn–Banach theorem. Bull. London Math. Soc. 42(4), 573–606 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  4. GW1

    Gowers W.T., Wolf J.: The true complexity of a system of linear equations. Proc. London Math. Soc. 100(3), 155–176 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  5. GW2

    W.T. Gowers, J. Wolf, Linear forms and quadratic uniformity for functions on \({\mathbb{F}^{n}_{p}}\), Arxiv preprint arXiv:1002.2209 (2010).

  6. GW3

    W.T. Gowers, J. Wolf, Linear forms and quadratic uniformity for functions on ZN, Arxiv preprint arXiv:1002.2210 (2010).

  7. Gr1

    Green B.J.: A Szemerédi-type regularity lemma in abelian groups. Geom. Funct. Anal. 15, 340–376 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  8. 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.

  9. GrT1

    Green B.J., Tao T.: An inverse theorem for the Gowers U 3(G) norm. Proc. Edinburgh Math. Soc. 51, 73–153 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  10. GrT2

    Green B.J., Tao T.: The primes contain arbitrarily long arithmetic progressions. Annals of Math. 167, 481–547 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  11. 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)

    MathSciNet  MATH  Google Scholar 

  12. GrT4

    Green B.J., Tao T.: Linear equations in primes. Annals of Math. 171, 1753–1850 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  13. 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).

  14. 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).

  15. L

    Leibman A.: Orbit of the diagonal of a power of a nilmanifold. Trans. Amer. Math. Soc. 362, 1619–1658 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  16. 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.

  17. 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.

  18. T

    T. Tao, A quantitative ergodic theory proof of Szemerédi’s theorem, Electron. J. Combin. 13:1 (2006), Research Paper 99 (2006).

    Google Scholar 

  19. TZ1

    Tao T., Ziegler T.: The primes contain arbitrarily long polynomial progressions. Acta Mathematica 201(2), 213–305 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  20. 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)

    MathSciNet  Article  Google Scholar 

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Correspondence to J. Wolf.

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Both authors gratefully acknowledge the hospitality of the Mathematical Sciences Research Institute, Berkeley, where important parts of this work were carried out.

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Gowers, W.T., Wolf, J. Linear Forms and Higher-Degree Uniformity for Functions On \({\mathbb{F}^{n}_{p}}\) . Geom. Funct. Anal. 21, 36–69 (2011). https://doi.org/10.1007/s00039-010-0106-3

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Keywords and phrases

  • Higher order Fourier analysis
  • uniformity norms
  • solutions to systems of linear equations

2010 Mathematics Subject Classification

  • 11B30