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Disjointness of the Möbius Transformation and Möbius Function

  • El Houcein El Abdalaoui
  • Igor E. ShparlinskiEmail author
Research
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Abstract

We study the distribution of the sequence of elements of the discrete dynamical system generated by the Möbius transformation \(x \mapsto (ax + b)/(cx + d)\) over a finite field of p elements. Motivated by a recent conjecture of P. Sarnak, we obtain nontrivial estimates of exponential sums with such sequences that imply that trajectories of this dynamical system are disjoined with the Möbius function.

Keywords

Möbius function Möbius transformation Möbius disjointness Exponential sums over primes 

Mathematics Subject Classification

11L07 11N60 11T23 37P05 

Notes

References

  1. 1.
    Baake, M., Neumärker, N., Roberts, J.A.G.: Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices. Discrete Cont. Dyn. Syst. Ser. A 33, 527–553 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Banks, W.D., Conflitti, A., Friedlander, J.B., Shparlinski, I.E.: Exponential sums over Mersenne numbers. Compos. Math. 140, 15–30 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Banks, W.D., Friedlander, J.B., Garaev, M.Z., Shparlinski, I.E.: Double character sums over elliptic curves and finite fields. Pure Appl. Math. Q. 2, 179–197 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Banks, W.D., Friedlander, J.B., Garaev, M.Z., Shparlinski, I.E.: Exponential and character sums Mersenne numbers. J. Aust. Math. Soc. 92, 1–13 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Blümel, R., Reinhardt, W.P.: Chaos in Atomic Physics. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  6. 6.
    Bourgain, J.: On the maximal ergodic theorem for certain subsets of the integers. Isr. J. Math. 61, 39–72 (1988)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bourgain, J.: On the pointwise ergodic theorem on \(L^p\) for arithmetic sets. Isr. J. Math. 61, 73–84 (1988)CrossRefGoogle Scholar
  8. 8.
    Bourgain, J.: An approach to pointwise ergodic theorems. In: Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math., vol. 1317, pp. 204–223. Springer, Berlin (1988)Google Scholar
  9. 9.
    Bourgain, J.: Pointwise ergodic theorems for arithmetic sets (with an appendix by J. Bourgain, H. Furstenberg, Y Katznelson and D.S. Ornstein). Inst. Hautes Études Sci. Publ. Math. 69, 5–45 (1989)CrossRefGoogle Scholar
  10. 10.
    Bourgain, J.: Estimates on exponential sums related to Diffie-Hellman distributions. Geom. Funct. Anal. 15, 1–34 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bourgain, J.: A remark on quantum ergodicity for CAT maps. In: Geometric Aspects of Functional Analysis, Lecture Notes in Math., vol. 1910, pp. 89–98. Springer, Berlin (2007)Google Scholar
  12. 12.
    Bourgain, J.: On the correlation of the Möbius function with rank-one systems. J. Anal. Math. 120, 105–130 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bourgain, J., Sarnak, P., Ziegler, T.: Disjointness of Möbius from horocycle flow. In: From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Devel. Math., vol. 28, pp. 67–83. Springer, New York (2013)Google Scholar
  14. 14.
    Buczolich, Z.: Ergodic averages with prime divisor weights in \(L^1\). Ergod. Theory Dyn. Syst. (2017).  https://doi.org/10.1017/etds.2017.54
  15. 15.
    Carmon, D., Rudnick, Z.: The autocorrelation of the Möbius function and Chowla’s conjecture for the rational function field. Q. J. Math. 65, 53–61 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chou, W.-S.: On inversive maximal period polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 6, 245–250 (1995)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cuny, C., Weber, M.: Ergodic theorems with arithmetical weights. Isr. J. Math. 217, 139–180 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Daboussi, H., Delange, H.: On multiplicative arithmetical functions whose modulus does not exceed one. J. Lond. Math. Soc. 26, 245–264 (1982)MathSciNetCrossRefGoogle Scholar
  19. 19.
    el Abdalaoui, E.H.: On Veech’s proof of Sarnak’s theorem on the Möbius flow. Preprint (2017). arXiv:1711.06326
  20. 20.
    el Abdalaoui, E.H., Lemańczyk, M., de la Rue, T.: On spectral disjointness of powers for rank-one transformations and Möbius orthogonality. J. Funct. Anal. 266, 284–317 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Eisner, T.: A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris 353, 569–572 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    el Abdalaoui, E.H., Kułaga-Przymus, J., Lemańczyk, M., de la Rue, T.: The Chowla and the Sarnak conjectures from ergodic theory point of view. Discrete Cont. Dyn. Syst. Ser. A 37, 2899–2944 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Eisner, T., Lin, M.: On modulated ergodic theorems. Preprint (2017). arXiv:1709.05322
  24. 24.
    Everest, G., van der Poorten, A., Shparlinski, I.E., Ward, T.: Recurrence sequences. In: Math. Surveys and Monogr., vol. 104. Amer. Math. Soc., Providence (2003)Google Scholar
  25. 25.
    Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M.: Sarnak’s conjecture—what’s new. In: Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, Lecture Notes in Mathematics, vol. 2213, pp. 163–235. Springer, Berlin (2018)Google Scholar
  26. 26.
    Fouvry, É., Ganguly, S.: Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms. Compos. Math. 150, 763–797 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Friedlander, J.B., Hansen, J., Shparlinski, I.E.: On character sums with exponential functions. Mathematika 47, 75–85 (2000)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Garaev, M.Z., Shparlinski, I.E.: The large sieve inequality with exponential functions and the distribution of Mersenne numbers modulo primes. Int. Math. Res. Not. 39, 2391–2408 (2005)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Gomilko, A., Kwietniak, D., Lemańczyk, M.: Sarnak’s conjecture implies the Chowla conjecture along a subsequence. In: Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, Lecture Notes in Math., vol. 2213, pp. 237–247. Springer, Cham (2018). arXiv:1710.07049
  30. 30.
    Green, B.J.: A note on multiplicative functions on progressions to large moduli. Proc. R. Soc. Edinb. 148A, 63–77 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Green, B.J., Tao, T.: The Möbius function is strongly orthogonal to nilsequences. Ann. Math. 175, 541–566 (2012)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory. Amer. Math. Soc., Providence (2004)zbMATHGoogle Scholar
  33. 33.
    Kátai, I.: A remark on a theorem of H. Daboussi. Acta Math. Hungar. 47, 223–225 (1986)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Kelmer, D.: On matrix elements for the quantized cat cap modulo prime powers. Ann. Henri Poincaré 9, 1479–1501 (2008)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kułaga-Przymus, J., Lemańczyk, M.: The Möbius function and continuous extensions of rotations. Monat. Math. 178, 553–582 (2015)CrossRefGoogle Scholar
  36. 36.
    Kurlberg, P.: Bounds on supremum norms for Hecke eigenfunctions. Ann. Henri Poincaré 8, 75–89 (2007)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Kurlberg, P., Rosenzweig, L., Rudnick, Z.: Matrix elements for the quantum cat map: fluctuations in short windows. Nonlinearity 20, 2289–2304 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103, 47–77 (2000)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Kurlberg, P., Rudnick, Z.: On the distribution of matrix elements for the quantum cat map. Ann. Math. 161, 489–507 (2005)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Li, W.-C.W.: Number Theory with Applications. World Scientific, Singapore (1996)CrossRefGoogle Scholar
  41. 41.
    Li, W.-C.W.: Character sums over \(p\)-adic fields. J. Number Theory 174, 181–229 (1999)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Li, W.-C.W.: Character sums over norm groups. Finite Fields Appl. 12, 1–15 (2006)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Liu, J., Sarnak, P.: The Möbius disjointness conjecture for distal flows. Duke Math. J. 164, 1353–1399 (2015)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Marchetti, D.H.U., Wreszinski, W.F.: Asymptotic Time Decay in Quantum Physics. World Scientific, Singapore (2013)CrossRefGoogle Scholar
  45. 45.
    Nair, R.: On polynomials in primes and J. Bourgain’s circle method approach to ergodic theorems. Ergod. Theory Dyn. Syst. 11, 485–499 (1991)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Nair, R.: On polynomials in primes and J. Bourgain’s circle method approach to ergodic theorems II. Studia Math. 105, 207–233 (1993)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Ostafe, A., Shparlinski, I.E.: Exponential sums over points of elliptic curves with reciprocals of primes. Mathematika 58, 21–33 (2012)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Ostafe, A., Shparlinski, I.E.: On the power generator of pseudorandom numbers and its multivariate analogue. J. Complexity 28, 238–249 (2012)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Ramaré, O.: Arithmetical aspects of the large sieve inequality (with the collaboration of D. S. Ramana). In: Harish–Chandra Res. Inst. Lecture Notes, vol. 1. Hindustan Book Agency, New Delhi (2009)Google Scholar
  50. 50.
    Rosenblatt, J.M., Wierdl, M.: Pointwise ergodic theorems via harmonic analysis. In: Ergodic Theory and Its Connections with Harmonic Analysis (Alexandria, 1993), London Math. Soc. Lecture Note Ser., vol. 205, pp. 3–151. Cambridge University Press, Cambridge (1995)Google Scholar
  51. 51.
    Rosenzweig, L.: Fluctuations of matrix elements of the quantum cat map. Int. Math. Res. Not. 2011, 4884–4933 (2011)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Ryzhikov, V.V.: Bounded ergodic constructions, disjointness, and weak limits of powers. Trans. Moscow Math. Soc. 74, 165–171 (2013)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Sarnak, P.: Möbius randomness and dynamics. Not. S. Afr. Math. Soc. 43, 89–97 (2012)MathSciNetGoogle Scholar
  54. 54.
    Sarnak, P., Ubis, A.: The horocycle flow at prime times. J. Math. Pures Appl. 103, 575–618 (2015)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Tao, T.: Equivalence of the logarithmically averaged Chowla and Sarnak conjectures. In: Elsholtz, C., Grabner, P. (eds.) Number Theory—Diophantine Problems, Uniform Distribution and Applications; Festschrift in Honour of Robert F. Tichy’s 60th Birthday, pp. 391–421. Springer, Berlin (2017)Google Scholar
  56. 56.
    Tao, T., Teräväinen, J.: The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures. Duke Math. J. (2017)Google Scholar
  57. 57.
    Thouvenot, J.-P.: La convergence presque sûre des moyennes ergodiques suivant certaines sous-suites d’entiers (d’après Jean Bourgain), Séminaire Bourbaki, vol. 1989/90. Astérisque 189–190, Exp. No. 719, pp. 133–153 (1990)Google Scholar
  58. 58.
    Weil, A.: Basic Number Theory. Springer, New York (1974)CrossRefGoogle Scholar
  59. 59.
    Wierdl, M.: Pointwise ergodic theorem along the prime numbers. Isr. J. Math. 64, 315–336 (1988)MathSciNetCrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Raphaël SalemUniversité de Rouen NormandieSaint-Étienne-du-RouvrayFrance
  2. 2.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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