Advertisement

Journal d'Analyse Mathématique

, Volume 135, Issue 2, pp 437–471 | Cite as

A continuous model for systems of complexity 2 on simple abelian groups

  • Pablo Candela
  • Balázs Szegedy
Article
  • 26 Downloads

Abstract

It is known that if p is a sufficiently large prime, then, for every function f: Zp → [0, 1], there exists a continuous function f′: T → [0, 1] on the circle such that the averages of f and f′ across any prescribed system of linear forms of complexity 1 differ by at most ∈. This result follows from work of Sisask, building on Fourier-analytic arguments of Croot that answered a question of Green. We generalize this result to systems of complexity at most 2, replacing T with the torus T2 equipped with a specific filtration. To this end, we use a notion of modelling for filtered nilmanifolds, that we define in terms of equidistributed maps and combine this notion with tools of quadratic Fourier analysis. Our results yield expressions on the torus for limits of combinatorial quantities involving systems of complexity 2 on Zp. For instance, let m4(α, Zp) denote the minimum, over all sets A ⊆ Zp of cardinality at least αp, of the density of 4-term arithmetic progressions inside A. We show that limp→∞ m4(α, Zp) is equal to the infimum, over all continuous functions f: T2 →[0, 1] with \({\smallint _{{T^2}}}f \geqslant a\), of the integral
$$\int_{{T^5}} {f\left( {\begin{array}{*{20}{c}} {{x_1}} \\ {{y_1}} \end{array}} \right)} f\left( {\begin{array}{*{20}{c}} {{x_1} + {x_2}} \\ {{y_1} + {y_2}} \end{array}} \right)f\left( {\begin{array}{*{20}{c}} {{x_1} + 2{x_2}} \\ {{y_1} + 2{y_2} + {y_3}} \end{array}} \right).f\left( {\begin{array}{*{20}{c}} {{x_1} + 3{x_2}} \\ {{y_1} + 3{y_2} = 3{y_3}} \end{array}} \right)d{\mu _{{T^5}}}({x_1},{x_2},{y_1},{y_2},{y_3})$$

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Candela and O. Sisask, On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime, Acta Math. Hungar. 132 (2011), 223–243.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    P. Candela and O. Sisask, Convergence results for systems of linear forms on cyclic groups, and periodic nilsequences, SIAM J. Discrete Math. 28 (2014), 786–810.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    E. Croot, The minimal number of three-term arithmetic progressions modulo a prime converges to a limit, Canad. Math. Bull. 51 (2008), 47–56.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    E. Croot and V. Lev, Open problems in additive combinatorics, Additive Combinatorics, American Mathematical Society, Providence RI, 2007, pp. 207–233.CrossRefzbMATHGoogle Scholar
  5. [5]
    L. Corwin and F. P. Greenleaf, Representations of nilpotent Lie groups and their applications, Part 1: Basic theory and examples, Cambridge University Press, Cambridge, 1990.zbMATHGoogle Scholar
  6. [6]
    A. Deitmar and S. Echterhoff, Principles of Harmonic Analysis, Springer, 2009.zbMATHGoogle Scholar
  7. [7]
    T. Eisner and T. Tao, Large values of the Gowers-Host-Kra seminorms, J. Anal. Math. 117 (2012), 133–186.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    V. V. Gorbatsevich, A. L. Onishchik and E. B. Vinberg, Foundations of Lie Theory and Lie transformation groups, Springer 1997.zbMATHGoogle Scholar
  9. [9]
    W. T. Gowers, A new proof of Szemerédi’s theorem, Geom./Funct. Anal. 11 (2001), 465–588.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    W. T. Gowers and J. Wolf, Linear functions and quadratic uniformity for functions on ZN, J. Anal. Math. 115 (2011), 121–186.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    W. T. Gowers and J. Wolf, The true complexity of a system of linear equations, Proc. Lond. Math. Soc. (3) 100 (2010), 155–176.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    B. Green and T. Tao, Quadratic uniformity of the Möbius function, Ann. Inst. Fourier (Grenoble) 58 (2008), 1863–1935.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), 465–540.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    B. Green and T. Tao, An arithmetic regularity lemma, an associated counting lemma, and applications, An Irregular Mind, Szemerédi is 70, Janos Bolyai Math. Soc., Budapest, 2010, pp. 261–334.zbMATHGoogle Scholar
  15. [15]
    B. Green, T. Tao, and T. Ziegler, An inverse theorem for the Gowers Us+1[N]-norm, Ann. of Math. 176 (2012), 1231–1372.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. École Normale Sup. (3) 71 (1954), 101–190.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    A. Leibman, Polynomial sequences in groups, Journal of Algebra 201 (1998), 189–206.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A. Leibman, Orbit of the diagonal in the power of a nilmanifold, Trans. Amer. Math. Soc. 362 (2010), 1619–1658.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Comb. Theory, Ser. B 96 (2006), 933–957.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    O. Sisask, Combinatorial properties of large subsets of abelian groups, Ph. D. Thesis, University of Bristol, 2009.Google Scholar
  21. [21]
    B. Szegedy, Limits of functions on groups, Trans. Amer. Math. Soc., to appear.Google Scholar
  22. [22]
    B. Szegedy, On higher order Fourier analysis, preprint. arXiv:1203. 2260.Google Scholar
  23. [23]
    T. Tao and V. Vu, Additive Combinatorics, Cambridge University Press, Cambridge, 2006.CrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Universidad Autónoma de MadridCiudad Universitaria de CantoblancoMadridEspaña
  2. 2.Alfréd Rényi Institute of MathematicsBudapestHungary

Personalised recommendations