Journal d'Analyse Mathématique

, Volume 134, Issue 1, pp 55–105 | Cite as

A structure theorem for multiplicative functions over the Gaussian integers and applications

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Abstract

We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U3-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x2y2 = n2 for some Gaussian integer n. The analog of this statement over Z remains open.

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References

  1. [1]
    V. Bergelson, Ergodic Theory and Diophantine Problems: Topics in Symbolic Dynamics and Applications, Cambridge Univ. Press, Cambridge, 1996, pp. 167–205.Google Scholar
  2. [2]
    V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden’s and Szemerédi theorems, J. Amer. Math. Soc. 9 (1996), 725–753.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    V. Bergelson and R. McCutcheon, Recurrence for semigroup actions and a non-commutative Schur theorem, Topological Dynamics and Applications, Amer. Math. Soc., Providence, RI, 1998, pp. 205–222.MATHGoogle Scholar
  4. [4]
    N. Frantzikinakis and B. Host, Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc. 30 (2017), 67–157.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    N. Frantzikinakis and B. Host, Uniformity of multiplicative functions and partition regularity of some quadratic equations, arXiv: 1303. 4329.Google Scholar
  6. [6]
    H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204–256.CrossRefMATHGoogle Scholar
  7. [7]
    T. Gowers, A new proof of Szemerédi theorem, Geom. Funct. Anal. 11 (2001), 465–588.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    T. Gowers, Decompositions, approximate structure, transference, and the Hahn-Banach theorem, Bull. London Math. Soc. 42 (2010), 573–606.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    T. Gowers and J. Wolf, Linear forms and quadratic uniformity for functions on ZN, J. Anal. Math. 115 (2011), 121–186.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. 167 (2008), 481–547.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    B. Green and T. Tao, An arithmetic regularity lemma, associated counting lemma, and applications, An Irregular Mind, Janos Bolyai Math. Soc., Budapest, 2010, pp. 261–334.MATHGoogle Scholar
  12. [12]
    B. Green and T. Tao, Linear equations in primes, Ann. of Math. (2) 171 (2010), 1753–1850.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    B. Green and T. Tao, The Mobius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), 541–566.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of. Math. (2) 175 (2012), 465–540.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    B. Green and T. Tao, On the quantitative distribution of polynomial nilsequences - erratum, Ann. of Math. (2) 179 (2014), 11751183.CrossRefMATHGoogle Scholar
  16. [16]
    B. Green, T. Tao, and T. Ziegler, An inverse theorem for the Gowers U s+1-norm, Ann. of Math. (2) 176 (2012), 1231–1372.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    G. H. Hardy and S. Ramanujan, Twelve Lectures on Subjects Suggested by his Life and Work, 3rd ed., Chelsea, New York, 1999, p. 67.Google Scholar
  18. [18]
    B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), 397–488.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    A. Khalfalah and E. Szemeredi, On the number of monochromatic solutions of x+y = z 2, Combin. Probab. Comput. 15 (2006), 213–227.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    J. Neukirch, Algebraic Number Theory, Springer, New York, 1999.CrossRefMATHGoogle Scholar
  21. [21]
    R. Rado, Studien zur Kombinatorik, Math. Z. 36 (1933), 424–470.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    A. Sarközy, On difference sets of integers. III, ActaMath. Acad. Sci. Hungar. 31 (1978), 355–386.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    B. Szegedy, On higher order Fourier analysis, arXiv: 1203. 2260.Google Scholar
  24. [24]
    T. Tao, A quantitative ergodic theory proof of Szemeredi’s theorem, Electron. J. Combin. (2) 13 (2006), Research Paper 99.MathSciNetMATHGoogle Scholar
  25. [25]
    P. Tchebichef, Mémoire sur les nombres premiers, J. Math. Pures Appl. 17 (1852), 366–390.Google Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

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