December  2020, 40(12): 6877-6918. doi: 10.3934/dcds.2020260

Automatic sequences are orthogonal to aperiodic multiplicative functions

1. 

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopin Street 12/18, 87-100 Toruń, Poland

2. 

Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstr. 8-10, 1040 Wien, Austria

* Corresponding author: Clemens Müllner

Received  August 2019 Revised  April 2020 Published  July 2020

Given a finite alphabet
$ \mathbb{A} $
and a primitive substitution
$ \theta: \mathbb{A}\to \mathbb{A}^\lambda $
(of constant length
$ \lambda $
), let
$ (X_\theta,S) $
denote the corresponding dynamical system, where
$ X_\theta $
is the closure of the orbit via the left shift
$ S $
of a fixed point of the natural extension of
$ \theta $
to a self-map of
$ \mathbb{A}^{ {\mathbb{Z}}} $
. The main result of the paper is that all continuous observables in
$ X_\theta $
are orthogonal to any bounded, aperiodic, multiplicative function
$ \boldsymbol{u}: {\mathbb{N}}\to {\mathbb{C}} $
, i.e.
$ \lim\limits_{N\to\infty}\frac1N\sum\limits_{n\leq N}f(S^nx) \boldsymbol{u}(n) = 0 $
for all
$ f\in C(X_\theta) $
and
$ x\in X_\theta $
. In particular, each primitive automatic sequence, that is, a sequence read by a primitive finite automaton, is orthogonal to any bounded, aperiodic, multiplicative function.
Citation: Mariusz Lemańczyk, Clemens Müllner. Automatic sequences are orthogonal to aperiodic multiplicative functions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6877-6918. doi: 10.3934/dcds.2020260
References:
[1]

H. El AbdalaouiJ. Kułaga-PrzymusM. Lemańczyk and T. de la Rue, Möbius disjointness for models of an ergodic system and beyond, Israel J. Math., 228 (2018), 707-751.  doi: 10.1007/s11856-018-1784-z.  Google Scholar

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J.-P. Allouche and L. Goldmakher, Mock characters and the Kronecker symbol, J. Number Theory, 192 (2018), 356-372.  doi: 10.1016/j.jnt.2018.04.022.  Google Scholar

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J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563.  Google Scholar

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V. BergelsonJ. Kułaga-PrzymusM. Lemańczyk and F. K. Richter, Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics, Ergodic Theory Dynam. Systems, 39 (2019), 2332-2383.  doi: 10.1017/etds.2017.130.  Google Scholar

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V. BergelsonJ. Kułaga-PrzymusM. Lemańczyk and F. Richter, A structure theorem for level sets of multiplicative functions and applications, International Math. Research Notices, 5 (2020), 1300-1345.  doi: 10.1093/imrn/rny040.  Google Scholar

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J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, From Fourier analysis and number theory to Radon transforms and geometry, Springer, New York, 2013, 67-83. doi: 10.1007/978-1-4614-4075-8_5.  Google Scholar

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A. Danilenko and M. Lemańczyk, Spectral multiplicities for ergodic flows, Discrete Contin. Dyn. Syst., 33 (2013), 4271-4289.  doi: 10.3934/dcds.2013.33.4271.  Google Scholar

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F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (1977/78), 221-239.  doi: 10.1007/BF00534241.  Google Scholar

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M. Drmota, Subsequences of automatic sequences and uniform distribution, in Uniform Distribution and Quasi-Monte Carlo Methods, Radon Series in Computational and Applied Mathematics, Vol. 15, De Gruyter, Berlin, 2014, 87-104.  Google Scholar

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J.-M. DeshouillersM. Drmota and C. Müllner, Automatic sequences generated by synchronizing automata fulfill the Sarnak conjecture, Studia Math., 231 (2015), 83-95.  doi: 10.4064/sm8479-2-2016.  Google Scholar

[11]

T. Downarowicz and J. Serafin, Almost full entropy subshifts uncorrelated to the Möbius function, Int. Math. Res. Not. IMRN, (2019), no. 11, 3459-3472. doi: 10.1093/imrn/rnx192.  Google Scholar

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S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk, Sarnak's conjecture: what's new, in Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Springer, Cham, 2018,163-235.  Google Scholar

[13]

S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk and C. Mauduit, Substitutions and Möbius disjointness, in Ergodic theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, Amer. Math. Soc., Providence, RI, 2016,151-174. doi: 10.1090/conm/678.  Google Scholar

[14]

N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002. doi: 10.1007/b13861.  Google Scholar

[15]

N. Frantzikinakis and B. Host, Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc., 30 (2017), 67-157.  doi: 10.1090/jams/857.  Google Scholar

[16]

N. Frantzikinaki and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math., 187 (2018), 869-931.  doi: 10.4007/annals.2018.187.3.6.  Google Scholar

[17]

N. Frantzikinakis and B. Host, Furstenberg systems of bounded multiplicative functions and applications, International Mathematics Research Notices, 2 (2020), 1073-7928.  doi: 10.1093/imrn/rnz037.  Google Scholar

[18]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar

[19]

E. Glasner, Ergodic theory via joinings, in Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[20]

A. Gomilko, D. Kwietniak and M. Lemańczyk, Sarnak's conjecture implies the Chowla conjecture along a subsequence, in Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 2213, Springer, Cham, 2018. doi: 10.1007/978-3-319-74908-2_12.  Google Scholar

[21]

J. L. Herning, Spectrum and Factors of Substitution Dynamical Systems, Ph.D dissertation, George Washington University, 2013, 93 pp.  Google Scholar

[22]

B. Host and F. Parreau, Homomorphismes entre systèmes dynamiques définis par substitutions, Ergodic Theory Dynam. Systems, 9 (1989), 469-477.  doi: 10.1017/S0143385700005113.  Google Scholar

[23]

I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.  doi: 10.1007/BF01949145.  Google Scholar

[24]

O. Klurman and P. Kurlberg, A note on multiplicative automatic sequences, II, Bulletin of the London Mathematical Society, 52 (2020), 185-188.  doi: 10.1112/blms.12318.  Google Scholar

[25]

J. Konieczny, Möbius orthogonality for $q$-multiplicative sequences, preprint, arXiv: 1808.06196. Google Scholar

[26]

J. Konieczny, On multiplicative automatic sequences, Bulletin of the London Mathematical Society, 52 (2020), 175-184.  doi: 10.1112/blms.12317.  Google Scholar

[27]

J. Kułaga-Przymus and M. Lemańczyk, The Möbius function and continuous extensions of rotations, Monatsh. Math., 178 (2015), 553-582.  doi: 10.1007/s00605-015-0808-6.  Google Scholar

[28]

E. Lehrer, Toplogical mixing and uniquely ergodic models, Israel J. Math., 57 (1987), 239-255.  doi: 10.1007/BF02772176.  Google Scholar

[29]

M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions, Compositio Math., 65 (1988), 241-263.   Google Scholar

[30]

M. Lemańczyk and M. K. Mentzen, Compact subgroups in the centralizers of natural factors of an ergodic group extension of a rotation determine all factors, Ergodic Theory Dynam. Systems, 10 (1990), 763-776.  doi: 10.1017/S0143385700005885.  Google Scholar

[31]

K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math., 183 (2016), 1015-1056.  doi: 10.4007/annals.2016.183.3.6.  Google Scholar

[32]

K. MatomäkiM. Radziwiłł and T. Tao, An averaged form of Chowla's conjecture, Algebra Number Theory, 9 (2015), 2167-2196.  doi: 10.2140/ant.2015.9.2167.  Google Scholar

[33]

C. Mauduit and J. Rivat, Sur un problème de Gelfond : La somme des chiffres des nombres premiers, Ann. of Math., 171 (2010), 1591-1646.  doi: 10.4007/annals.2010.171.1591.  Google Scholar

[34]

C. Mauduit and J. Rivat, Prime numbers along Rudin-Shapiro sequences, J. Eur. Math. Soc., 17 (2015), 2595-2642.  doi: 10.4171/JEMS/566.  Google Scholar

[35]

M. K. Mentzen, Invariant sub-$\sigma$-algebras for substitutions of constant length, Studia Math., 92 (1989), 257-273.  doi: 10.4064/sm-92-3-257-273.  Google Scholar

[36]

M. K. Mentzen, Ergodic properties of group extensions of dynamical systems with discrete spectra, Studia Math., 101 (1991), 19-31.  doi: 10.4064/sm-101-1-19-31.  Google Scholar

[37]

C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219-3290.  doi: 10.1215/00127094-2017-0024.  Google Scholar

[38]

M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, second edition, Lecture Notes in Mathematics, Vol. 1294, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6.  Google Scholar

[39]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sites/default/files/Mobius%20lectures%20Summer%202010.pdf. Google Scholar

[40]

J.-C. Schlage-Puchta, Completely multiplicative automatic functions, Integers, 11 (2011), A31, 8 pp. doi: 10.1515/INTEG.2011.055.  Google Scholar

[41]

T. Tao, The logarithmically averaged and non logarithmically averaged Chowla conjectures, https://terrytao.wordpress.com/2017/10/20/the-logarithmically-averaged-and-non-logarithmically-averaged-chowla-conjectures/ Google Scholar

[42]

W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341.  doi: 10.1007/BF01667386.  Google Scholar

[43]

S. Yazdani, Multiplicative functions and k-automatic sequences, J. Théor. Nombres Bordeaux, 13 (2001), 651-658.  doi: 10.5802/jtnb.342.  Google Scholar

show all references

References:
[1]

H. El AbdalaouiJ. Kułaga-PrzymusM. Lemańczyk and T. de la Rue, Möbius disjointness for models of an ergodic system and beyond, Israel J. Math., 228 (2018), 707-751.  doi: 10.1007/s11856-018-1784-z.  Google Scholar

[2]

J.-P. Allouche and L. Goldmakher, Mock characters and the Kronecker symbol, J. Number Theory, 192 (2018), 356-372.  doi: 10.1016/j.jnt.2018.04.022.  Google Scholar

[3]

J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563.  Google Scholar

[4]

V. BergelsonJ. Kułaga-PrzymusM. Lemańczyk and F. K. Richter, Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics, Ergodic Theory Dynam. Systems, 39 (2019), 2332-2383.  doi: 10.1017/etds.2017.130.  Google Scholar

[5]

V. BergelsonJ. Kułaga-PrzymusM. Lemańczyk and F. Richter, A structure theorem for level sets of multiplicative functions and applications, International Math. Research Notices, 5 (2020), 1300-1345.  doi: 10.1093/imrn/rny040.  Google Scholar

[6]

J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, From Fourier analysis and number theory to Radon transforms and geometry, Springer, New York, 2013, 67-83. doi: 10.1007/978-1-4614-4075-8_5.  Google Scholar

[7]

A. Danilenko and M. Lemańczyk, Spectral multiplicities for ergodic flows, Discrete Contin. Dyn. Syst., 33 (2013), 4271-4289.  doi: 10.3934/dcds.2013.33.4271.  Google Scholar

[8]

F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 41 (1977/78), 221-239.  doi: 10.1007/BF00534241.  Google Scholar

[9]

M. Drmota, Subsequences of automatic sequences and uniform distribution, in Uniform Distribution and Quasi-Monte Carlo Methods, Radon Series in Computational and Applied Mathematics, Vol. 15, De Gruyter, Berlin, 2014, 87-104.  Google Scholar

[10]

J.-M. DeshouillersM. Drmota and C. Müllner, Automatic sequences generated by synchronizing automata fulfill the Sarnak conjecture, Studia Math., 231 (2015), 83-95.  doi: 10.4064/sm8479-2-2016.  Google Scholar

[11]

T. Downarowicz and J. Serafin, Almost full entropy subshifts uncorrelated to the Möbius function, Int. Math. Res. Not. IMRN, (2019), no. 11, 3459-3472. doi: 10.1093/imrn/rnx192.  Google Scholar

[12]

S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk, Sarnak's conjecture: what's new, in Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Springer, Cham, 2018,163-235.  Google Scholar

[13]

S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk and C. Mauduit, Substitutions and Möbius disjointness, in Ergodic theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, Amer. Math. Soc., Providence, RI, 2016,151-174. doi: 10.1090/conm/678.  Google Scholar

[14]

N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002. doi: 10.1007/b13861.  Google Scholar

[15]

N. Frantzikinakis and B. Host, Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc., 30 (2017), 67-157.  doi: 10.1090/jams/857.  Google Scholar

[16]

N. Frantzikinaki and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math., 187 (2018), 869-931.  doi: 10.4007/annals.2018.187.3.6.  Google Scholar

[17]

N. Frantzikinakis and B. Host, Furstenberg systems of bounded multiplicative functions and applications, International Mathematics Research Notices, 2 (2020), 1073-7928.  doi: 10.1093/imrn/rnz037.  Google Scholar

[18]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar

[19]

E. Glasner, Ergodic theory via joinings, in Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[20]

A. Gomilko, D. Kwietniak and M. Lemańczyk, Sarnak's conjecture implies the Chowla conjecture along a subsequence, in Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 2213, Springer, Cham, 2018. doi: 10.1007/978-3-319-74908-2_12.  Google Scholar

[21]

J. L. Herning, Spectrum and Factors of Substitution Dynamical Systems, Ph.D dissertation, George Washington University, 2013, 93 pp.  Google Scholar

[22]

B. Host and F. Parreau, Homomorphismes entre systèmes dynamiques définis par substitutions, Ergodic Theory Dynam. Systems, 9 (1989), 469-477.  doi: 10.1017/S0143385700005113.  Google Scholar

[23]

I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.  doi: 10.1007/BF01949145.  Google Scholar

[24]

O. Klurman and P. Kurlberg, A note on multiplicative automatic sequences, II, Bulletin of the London Mathematical Society, 52 (2020), 185-188.  doi: 10.1112/blms.12318.  Google Scholar

[25]

J. Konieczny, Möbius orthogonality for $q$-multiplicative sequences, preprint, arXiv: 1808.06196. Google Scholar

[26]

J. Konieczny, On multiplicative automatic sequences, Bulletin of the London Mathematical Society, 52 (2020), 175-184.  doi: 10.1112/blms.12317.  Google Scholar

[27]

J. Kułaga-Przymus and M. Lemańczyk, The Möbius function and continuous extensions of rotations, Monatsh. Math., 178 (2015), 553-582.  doi: 10.1007/s00605-015-0808-6.  Google Scholar

[28]

E. Lehrer, Toplogical mixing and uniquely ergodic models, Israel J. Math., 57 (1987), 239-255.  doi: 10.1007/BF02772176.  Google Scholar

[29]

M. Lemańczyk and M. K. Mentzen, On metric properties of substitutions, Compositio Math., 65 (1988), 241-263.   Google Scholar

[30]

M. Lemańczyk and M. K. Mentzen, Compact subgroups in the centralizers of natural factors of an ergodic group extension of a rotation determine all factors, Ergodic Theory Dynam. Systems, 10 (1990), 763-776.  doi: 10.1017/S0143385700005885.  Google Scholar

[31]

K. Matomäki and M. Radziwiłł, Multiplicative functions in short intervals, Ann. of Math., 183 (2016), 1015-1056.  doi: 10.4007/annals.2016.183.3.6.  Google Scholar

[32]

K. MatomäkiM. Radziwiłł and T. Tao, An averaged form of Chowla's conjecture, Algebra Number Theory, 9 (2015), 2167-2196.  doi: 10.2140/ant.2015.9.2167.  Google Scholar

[33]

C. Mauduit and J. Rivat, Sur un problème de Gelfond : La somme des chiffres des nombres premiers, Ann. of Math., 171 (2010), 1591-1646.  doi: 10.4007/annals.2010.171.1591.  Google Scholar

[34]

C. Mauduit and J. Rivat, Prime numbers along Rudin-Shapiro sequences, J. Eur. Math. Soc., 17 (2015), 2595-2642.  doi: 10.4171/JEMS/566.  Google Scholar

[35]

M. K. Mentzen, Invariant sub-$\sigma$-algebras for substitutions of constant length, Studia Math., 92 (1989), 257-273.  doi: 10.4064/sm-92-3-257-273.  Google Scholar

[36]

M. K. Mentzen, Ergodic properties of group extensions of dynamical systems with discrete spectra, Studia Math., 101 (1991), 19-31.  doi: 10.4064/sm-101-1-19-31.  Google Scholar

[37]

C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219-3290.  doi: 10.1215/00127094-2017-0024.  Google Scholar

[38]

M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, second edition, Lecture Notes in Mathematics, Vol. 1294, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6.  Google Scholar

[39]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sites/default/files/Mobius%20lectures%20Summer%202010.pdf. Google Scholar

[40]

J.-C. Schlage-Puchta, Completely multiplicative automatic functions, Integers, 11 (2011), A31, 8 pp. doi: 10.1515/INTEG.2011.055.  Google Scholar

[41]

T. Tao, The logarithmically averaged and non logarithmically averaged Chowla conjectures, https://terrytao.wordpress.com/2017/10/20/the-logarithmically-averaged-and-non-logarithmically-averaged-chowla-conjectures/ Google Scholar

[42]

W. A. Veech, A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341.  doi: 10.1007/BF01667386.  Google Scholar

[43]

S. Yazdani, Multiplicative functions and k-automatic sequences, J. Théor. Nombres Bordeaux, 13 (2001), 651-658.  doi: 10.5802/jtnb.342.  Google Scholar

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