Advanced Search
Article Contents
Article Contents

Möbius disjointness for skew products on a circle and a nilmanifold

  • * Corresponding author: Ke Wang

    * Corresponding author: Ke Wang
Abstract Full Text(HTML) Related Papers Cited by
  • Let $ \mathbb{T} $ be the unit circle and $ \Gamma \backslash G $ the $ 3 $-dimensional Heisenberg nilmanifold. We prove that a class of skew products on $ \mathbb{T} \times \Gamma \backslash G $ are distal, and that the Möbius function is linearly disjoint from these skew products. This verifies the Möbius Disjointness Conjecture of Sarnak.

    Mathematics Subject Classification: 37A44, 11L03, 11N37.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] J. Bourgain, On the correlation of the Möbius function with rank-one systems, J. Anal. Math., 120 (2013), 105-130.  doi: 10.1007/s11854-013-0016-z.
    [2] J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., vol. 28, Springer, New York, 2013, 67–83. doi: 10.1007/978-1-4614-4075-8_5.
    [3] H. Davenport, On some infinite series involving arithmetical functions, II, Quart. J. Math., 8 (1937), 313-350.  doi: 10.1093/qmath/os-8.1.313.
    [4] A. de Faveri, Möbius disjiontness for $C^{1+\epsilon}$ skew products, preprint, arXiv: 2002.01076.
    [5] A.-H. Fan and Y. Jiang, Oscillating sequences, MMA and MMLS flows and Sarnak's conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 1709-1744.  doi: 10.1017/etds.2016.121.
    [6] S. Ferenczi, J. Kulaga-Przymus and M. Lemanczyk, Sarnak's conjecture: What's new, in Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, Lecture Notes in Math., vol. 2213, Springer, Cham, 2018,163–235.
    [7] H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601.  doi: 10.2307/2372899.
    [8] H. Furstenberg, The structure of distal flows, Amer. J. Math., 85 (1963), 477-515.  doi: 10.2307/2373137.
    [9] B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.
    [10] B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math., 175 (2012), 541-566.  doi: 10.4007/annals.2012.175.2.3.
    [11] L. K. Hua, Additive theory of prime numbers, Transl. Math. Monogr. 13, Amer. Math. Soc., Providence, 1965.
    [12] W. HuangZ. Wang and X. Ye, Measure complexity and Möbius disjointness, Adv. Math., 347 (2019), 827-858.  doi: 10.1016/j.aim.2019.03.007.
    [13] W. HuangZ. Wang and G. Zhang, Möbius disjointness for topological model of any ergodic system with discrete spectrum, J. Mod. Dyn., 14 (2019), 277-290.  doi: 10.3934/jmd.2019010.
    [14] A. Kanigowski, M. Lemanczyk and M. Radziwill, Rigidity in dynamics and Möbius disjointness, preprint, arXiv: 1905.13256v2.
    [15] M. Litman and Z. Wang, Möbius disjointness for skew products on the Heisenberg nilmanifold, Proc. Amer. Math. Soc., 147 (2019), 2033-2043.  doi: 10.1090/proc/14259.
    [16] J. Liu and P. Sarnak, The Möbius function and distal flows, Duke Math. J., 164 (2015), 1353-1399.  doi: 10.1215/00127094-2916213.
    [17] J. Liu and P. Sarnak, The Möbius disjointness conjecture for distal flows, in Proceedings of the Sixth International Congress of Chinese Mathematician, Vol. I, Adv. Lect. Math. (ALM) 36, Int. Press, Somerville, MA, 2017, 327-335.
    [18] A. I. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9-32. 
    [19] K. MatomäkiM. Radziwill and T. Tao, An averaged form of Chowla's conjecture, Algebra Number Theory, 9 (2015), 2167-2196.  doi: 10.2140/ant.2015.9.2167.
    [20] W. Parry, Zero entropy of distal and related transformations, Topological Dynamics, (Symposium, Colorado State Univ., Ft. Collins, Colo., 1967), 383–389.
    [21] R. Peckner, Möbius disjointness for homogeneous dynamics, Duke Math. J., 167 (2018), 2745-2792.  doi: 10.1215/00127094-2018-0026.
    [22] P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, IAS Lecture Notes, 2009; https://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf.
    [23] P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97. 
    [24] R. Tolimieri, Analysis on the Heisenberg manifold, Trans. Amer. Math. Soc., 288 (1977), 329-343.  doi: 10.2307/1998533.
    [25] Z. Wang, Möbius disjointness for analytic skew products, Invent. Math., 209 (2017), 175-196.  doi: 10.1007/s00222-016-0707-z.
  • 加载中

Article Metrics

HTML views(500) PDF downloads(265) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint