Möbius disjointness for skew products on the Heisenberg nilmanifold

Litman, Matthew; Wang, Zhiren

doi:10.1090/proc/14259

Möbius disjointness for skew products on the Heisenberg nilmanifold
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by Matthew Litman and Zhiren Wang PDF

Proc. Amer. Math. Soc. 147 (2019), 2033-2043 Request permission

Abstract:

We prove that the Möbius function is disjoint to all Lipschitz continuous skew product dynamical systems on the 3-dimensional Heisenberg nilmanifold over a minimal rotation of the 2-dimensional torus.

References

J. Bourgain, P. Sarnak, and T. Ziegler, Disjointness of Moebius from horocycle flows, From Fourier analysis and number theory to Radon transforms and geometry, Dev. Math., vol. 28, Springer, New York, 2013, pp. 67–83. MR 2986954, DOI 10.1007/978-1-4614-4075-8_{5}
H. Davenport, On some infinite series involving arithmetical functions II, Quat. J. Math. 8 (1937), 313–320.
Robert Ellis, Shmuel Glasner, and Leonard Shapiro, Proximal-isometric ($\scr P\scr J$) flows, Advances in Math. 17 (1975), no. 3, 213–260. MR 380755, DOI 10.1016/0001-8708(75)90093-6
Sébastien Ferenczi, Joanna Kułaga-Przymus, and Mariusz Lemańczyk, Sarnak’s Conjecture – what’s new, preprint (2017), arXiv:1710.04039.
Livio Flaminio, Krzyszto Fraczek, Joanna Kułaga-Przymus, and Mariusz Lemańczyk, Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds , Studia Math. (to appear).
Nikos Frantzikinakis, Ergodicity of the Liouville system implies the Chowla conjecture, Discrete Anal. , posted on (2017), Paper No. 19, 41. MR 3742396, DOI 10.19086/da.2733
Nikos Frantzikinakis and Bernard Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math. (2) 187 (2018), no. 3, 869–931. MR 3779960, DOI 10.4007/annals.2018.187.3.6
H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math. 83 (1961), 573–601. MR 133429, DOI 10.2307/2372899
H. Furstenberg, The structure of distal flows, Amer. J. Math. 85 (1963), 477–515. MR 157368, DOI 10.2307/2373137
Ben Green and Terence Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), no. 2, 465–540. MR 2877065, DOI 10.4007/annals.2012.175.2.2
Ben Green and Terence Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), no. 2, 541–566. MR 2877066, DOI 10.4007/annals.2012.175.2.3
I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar. 47 (1986), no. 1-2, 223–225. MR 836415, DOI 10.1007/BF01949145
J. Kułaga-Przymus and M. Lemańczyk, The Möbius function and continuous extensions of rotations, Monatsh. Math. 178 (2015), no. 4, 553–582. MR 3422903, DOI 10.1007/s00605-015-0808-6
Jianya Liu and Peter Sarnak, The Möbius function and distal flows, Duke Math. J. 164 (2015), no. 7, 1353–1399. MR 3347317, DOI 10.1215/00127094-2916213
Kaisa Matomäki, Maksym Radziwiłł, and Terence Tao, An averaged form of Chowla’s conjecture, Algebra Number Theory 9 (2015), no. 9, 2167–2196. MR 3435814, DOI 10.2140/ant.2015.9.2167
Peter Sarnak, Three lectures on the Möbius function, randomness and dynamics, lecture notes, IAS (2009).
Terence Tao, Equivalence of the logarithmically averaged Chowla and Sarnak conjectures, Number theory—Diophantine problems, uniform distribution and applications, Springer, Cham, 2017, pp. 391–421. MR 3676413
William A. Veech, Topological dynamics, Bull. Amer. Math. Soc. 83 (1977), no. 5, 775–830. MR 467705, DOI 10.1090/S0002-9904-1977-14319-X
Zhiren Wang, Möbius disjointness for analytic skew products, Invent. Math. 209 (2017), no. 1, 175–196. MR 3660308, DOI 10.1007/s00222-016-0707-z