Duke Mathematical Journal

The Möbius function and distal flows

Jianya Liu and Peter Sarnak

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We prove that the Möbius function is linearly disjoint from an analytic skew product on the 2 -torus. These flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points. The previous cases for which such disjointness has been proved are all regular. We also establish the linear disjointness of the Möbius function from various distal homogeneous flows.

Article information

Duke Math. J., Volume 164, Number 7 (2015), 1353-1399.

Received: 27 June 2013
Revised: 28 June 2014
First available in Project Euclid: 14 May 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11L03: Trigonometric and exponential sums, general
Secondary: 37A45: Relations with number theory and harmonic analysis [See also 11Kxx] 11N37: Asymptotic results on arithmetic functions

the Möbius function distal flow affine linear map skew product nilmanifold


Liu, Jianya; Sarnak, Peter. The Möbius function and distal flows. Duke Math. J. 164 (2015), no. 7, 1353--1399. doi:10.1215/00127094-2916213. https://projecteuclid.org/euclid.dmj/1431608070

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  • [1] N. Aoki, Topological entropy of distal affine transformations on compact abelian groups, J. Math. Soc. Japan 23 (1971), 11–17.
  • [2] J. Bourgain, On the correlation of the Moebius function with random rank-one systems, J. Anal. Math. 120 (2013), 105–130.
  • [3] J. Bourgain, P. Sarnak, and T. Ziegler, “Disjointness of Moebius from horocycle flows” in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math. 28, Springer, New York, 2013, 67–83.
  • [4] S. G. Dani, “Dynamical systems on homogeneous spaces” in Dynamical Systems, Ergodic Theory and Applications, Encyclopaedia Math. Sci. 100, Springer, Berlin, 266–359.
  • [5] H. Davenport, On some infinite series involving arithmetical functions, II, Quart. J. Math. 8 (1937), 313–350.
  • [6] N. M. dos Santos and R. Urzúa-Luz, Minimal homeomorphisms on low-dimensional tori, Ergodic Theory Dynam. Systems 29 (2009), 1515–1528.
  • [7] H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math. 83 (1961), 573–601.
  • [8] H. Furstenberg, The structure of distal flows, Amer. J. Math. 85 (1963), 477–515.
  • [9] B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), 541–566.
  • [10] B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), 465–540.
  • [11] F. J. Hahn, On affine transformations of compact abelian groups, Amer. J. Math. 85 (1963), 428–446. Errata, Amer. J. Math. 86 (1964), 463–464.
  • [12] H. Hoare and W. Parry, Affine transformations with quasi-discrete spectrum, I, J. London Math. Soc. 41 (1966), 88–96.
  • [13] L. K. Hua, Additive Theory of Prime Numbers, Transl. Math. Monogr. 13, Amer. Math. Soc., Providence, 1965.
  • [14] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, Amer. Math. Soc., Providence, 2004.
  • [15] I. Kátai, A remark on a theorem of Daboussi, Acta Math. Hungar. 47 (1986), 223–225.
  • [16] A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), 201–213.
  • [17] J. Liu and P. Sarnak, “The Möbius disjointness conjecture for distal flows” in Proceedings of the Sixth International Congress of Chinese Mathematicians, to appear, preprint, arXiv:1406.7243v1 [math.NT].
  • [18] C. Mauduit and J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. of Math. (2) 171 (2010), 1591–1646.
  • [19] P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, preprint, http://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf (accessed 12 January 2010).
  • [20] P. Sarnak, Mobius randomness and dynamics, Not. S. Afr. Math. Soc. 43 (2012), 89–97.
  • [21] P. Sarnak and A. Ubis, The horocycle at prime times, J. Math. Pures Appl. (9) 103 (2015), 575–618.