Sarnak’s conjecture for a class of rank-one subshifts

Etedadialiabadi, Mahmood; Gao, Su

doi:10.1090/bproc/148

Sarnak’s conjecture for a class of rank-one subshifts
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by Mahmood Etedadialiabadi and Su Gao HTML | PDF

Proc. Amer. Math. Soc. Ser. B 9 (2022), 460-471

Abstract:

Using techniques developed by Kanigowski, Lemańczyk, and Radziwiłł [Fund. Math. 255 (2021), pp. 309–336], we verify Sarnak’s conjecture for two classes of rank-one subshifts with unbounded cutting parameters. The first class of rank-one subshifts we consider is called almost complete congruency classes (accc), the definition of which is motivated by the main result of Foreman, Gao, Hill, Silva, and Weiss [Isr. J. Math., To appear], which implies that when a rank-one subshift carries a unique nonatomic invariant probability measure, it is accc if it is measure-theoretically isomorphic to an odometer. The second class we consider consists of Katok’s map and its generalizations.

References

El Houcein El Abdalaoui, Joanna Kułaga-Przymus, Mariusz Lemańczyk, and Thierry de la Rue, The Chowla and the Sarnak conjectures from ergodic theory point of view, Discrete Contin. Dyn. Syst. 37 (2017), no. 6, 2899–2944. MR 3622068, DOI 10.3934/dcds.2017125
El Houcein El Abdalaoui, Mariusz Lemańczyk, and Thierry de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal. 266 (2014), no. 1, 284–317. MR 3121731, DOI 10.1016/j.jfa.2013.09.005
Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
J. Bourgain, On the correlation of the Moebius function with rank-one systems, J. Anal. Math. 120 (2013), 105–130. MR 3095150, DOI 10.1007/s11854-013-0016-z
Tomasz Downarowicz and Stanisław Kasjan, Odometers and Toeplitz systems revisited in the context of Sarnak’s conjecture, Studia Math. 229 (2015), no. 1, 45–72. MR 3459905, DOI 10.4064/sm8314-12-2015
Sébastien Ferenczi, Systems of finite rank, Colloq. Math. 73 (1997), no. 1, 35–65. MR 1436950, DOI 10.4064/cm-73-1-35-65
Sébastien Ferenczi, Joanna Kułaga-Przymus, and Mariusz Lemańczyk, Sarnak’s conjecture: what’s new, Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics, Lecture Notes in Math., vol. 2213, Springer, Cham, 2018, pp. 163–235. MR 3821717
Sébastien Ferenczi and Christian Mauduit, On Sarnak’s conjecture and Veech’s question for interval exchanges, J. Anal. Math. 134 (2018), no. 2, 545–573. MR 3771491, DOI 10.1007/s11854-018-0017-z
M. Foreman, S. Gao, A. Hill, C. E. Silva, and B. Weiss, Rank-one transformations, odometers, and finite factors, Israel J. Math., To appear.
Wen Huang, Zhiren Wang, and Xiangdong Ye, Measure complexity and Möbius disjointness, Adv. Math. 347 (2019), 827–858. MR 3920840, DOI 10.1016/j.aim.2019.03.007
W. Huang, Z. Lian, S. Shao, and X. Ye, Reducing the Sarnak conjecture to Toeplitz systems, arXiv:1908.07554, 2019.
Adam Kanigowski, Mariusz Lemańczyk, and Maksym Radziwiłł, Rigidity in dynamics and Möbius disjointness, Fund. Math. 255 (2021), no. 3, 309–336. MR 4324828, DOI 10.4064/fm931-11-2020
A. Kanigowski, M. Lemańczyk, and M. Radziwiłł, Prime number theorem for analytic skew products, arXiv:2004.01125, 2020.
T. Tao, The Chowla conjecture and the Sarnak conjecture, https://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-the-sarnak-conjecture/.
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550