Duke Mathematical Journal

Automatic sequences fulfill the Sarnak conjecture

Clemens Müllner

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We present in this article a new method for dealing with automatic sequences. This method allows us to prove a Möbius randomness principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences. Furthermore, we can show a prime number theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to 0.

Article information

Duke Math. J., Volume 166, Number 17 (2017), 3219-3290.

Received: 4 May 2016
Revised: 21 April 2017
First available in Project Euclid: 5 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A63: Radix representation; digital problems {For metric results, see 11K16}
Secondary: 11B85: Automata sequences 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx] 11N05: Distribution of primes 11L20: Sums over primes

automata sequence Sarnak conjecture symbolic dynamics sums over primes exponential sums


Müllner, Clemens. Automatic sequences fulfill the Sarnak conjecture. Duke Math. J. 166 (2017), no. 17, 3219--3290. doi:10.1215/00127094-2017-0024. https://projecteuclid.org/euclid.dmj/1507169019

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  • [1] J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge Univ. Press, Cambridge, 2003.
  • [2] J. Bourgain, Möbius-Walsh correlation bounds and an estimate of Mauduit and Rivat, J. Anal. Math. 119 (2013), 147–163.
  • [3] J. Bourgain, On the correlation of the Moebius function with rank-one systems, J. Anal. Math. 120 (2013), 105–130.
  • [4] 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.
  • [5] C. Dartyge and G. Tenenbaum, Sommes des chiffres de multiples d’entiers, Ann. Inst. Fourier (Grenoble) 55 (2005), 2423–2474.
  • [6] H. Davenport, On some infinite series involving arithmetical functions, II, Quart. J. Math., Oxf. Ser. 8 (1937), 313–320.
  • [7] J.-M. Deshouillers, M. Drmota, and C. Müllner. Automatic sequences generated by synchronizing automata fulfill the Sarnak conjecture, Studia Math. 231 (2015), 83–95.
  • [8] T. Downarowicz and S. Kasjan, Odometers and Toeplitz systems revisited in the context of Sarnak’s conjecture, Studia Math. 229 (2015), 45–72.
  • [9] M. Drmota, “Subsequences of automatic sequences and uniform distribution” in Uniform Distribution and Quasi-Monte Carlo Methods, Radon Ser. Comput. Appl. Math. 15, De Gruyter, Berlin, 2014, 87–104.
  • [10] M. Drmota and J. F. Morgenbesser, Generalized Thue–Morse sequences of squares, Israel J. Math. 190 (2012), 157–193.
  • [11] E. H. El Abdalaoui, S. Kasjan, and M. Lemańczyk, $0$–$1$ sequences of the Thue–Morse type and Sarnak’s conjecture, Proc. Amer. Math. Soc. 144 (2016), 161–176.
  • [12] E. H. El Abdalaoui, J. Kułaga-Przymus, M. Lemańczyk, and T. De La Rue, The Chowla and the Sarnak conjectures from ergodic theory point of view, Discrete Contin. Dyn. Syst. 37 (2017), 2899–2944.
  • [13] E. H. El Abdalaoui, M. Lemańczyk, and T. de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal. 266 (2014), 284–317.
  • [14] E. H. El Abdalaoui, M. Lemańczyk, and T. de la Rue, Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals, Int. Math. Res. Not. IMRN 2017, no. 14, 4350–4368.
  • [15] S. Ferenczi, Complexity of sequences and dynamical systems, Discrete Math. 206 (1999), 145–154.
  • [16] S. Ferenczi, J. Kułaga-Przymus, M. Lemanczyk, and C. Mauduit, “Substitutions and Möbius disjointness” in Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, Contemp. Math. 678, Amer. Math. Soc., Providence, 2016, 151–173.
  • [17] S. Ferenczi and C. Mauduit, On Sarnak’s conjecture and Veech’s question for interval exchanges, preprint, http://iml.univ-mrs.fr/~ferenczi/fm2.pdf (accessed 16 February 2017).
  • [18] N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Math. 1794, Springer, Berlin, 2002.
  • [19] N. Frantzikinakis, Ergodicity of the Liouville system implies the Chowla conjecture, preprint, arXiv:1611.09338v2 [math.NT].
  • [20] S. W. Graham and G. Kolesnik, van der Corput’s Method of Exponential Sums, London Math. Soc. Lecture Note Ser. 126, Cambridge Univ. Press, Cambridge, 1991.
  • [21] B. Green, On (not) computing the Möbius function using bounded depth circuits, Combin. Probab. Comput. 21 (2012), 942–951.
  • [22] B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175 (2012), 541–566.
  • [23] G. Hanna, Sur les occurrences des mots dans les nombres premiers, Acta Arith. 178 (2017), 15–42.
  • [24] K.-H. Indlekofer and I. Kátai, Investigations in the theory of $q$-additive and $q$-multiplicative functions, I, Acta Math. Hungar. 91 (2001), 53–78.
  • [25] D. Karagulyan, On Möbius orthogonality for interval maps of zero entropy and orientation-preserving circle homeomorphisms, Ark. Mat. 53 (2015), 317–327.
  • [26] I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar. 47 (1986), 223–225.
  • [27] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure Appl. Math., Wiley-Interscience, New York, 1974.
  • [28] J. Kułaga Przymus and M. Lemańczyk, The Möbius function and continuous extensions of rotations, Monatsh. Math. 178 (2015), 553–582.
  • [29] J. Liu and P. Sarnak, The Möbius function and distal flows, Duke Math. J. 164 (2015), 1353–1399.
  • [30] 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.
  • [31] C. Mauduit and J. Rivat, Prime numbers along Rudin–Shapiro sequences, J. Eur. Math. Soc. (JEMS) 17 (2015), 2595–2642.
  • [32] R. Peckner, Möbius disjointness for homogeneous dynamics, preprint, arXiv:1506.07778v2 [math.NT].
  • [33] P. Sarnak, Three lectures on the Mobius function randomness and dynamics, preprint, https://www.math.ias.edu/files/wam/2011/PSMobius.pdf (accessed 20 March 2015).
  • [34] P. Sarnak and A. Ubis, The horocycle flow at prime times, J. Math. Pures Appl. (9) 103 (2015), 575–618.
  • [35] J.-P. Serre, Linear Representations of Finite Groups, Grad. Texts in Math. 42, Springer, New York, 1977.
  • [36] T. Tao, Möbius randomness of the Rudin–Shapiro sequence, http://mathoverflow.net/questions/97261 (accessed 7 September 2015).
  • [37] T. Tao, “Equivalence of the logarithmically averaged Chowla and Sarnak conjectures” in Number Theory—Diophantine Problems, Uniform Distribution and Applications, Springer, Cham, 2017, 391–421.
  • [38] W. A. Veech, Möbius orthogonality for generalized Morse–Kakutani flows, to appear in Amer. J. Math.