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Maximal equicontinuous generic factors and weak model sets
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 |
$ \mathbb{A} $ |
$ \theta: \mathbb{A}\to \mathbb{A}^\lambda $ |
$ \lambda $ |
$ (X_\theta,S) $ |
$ X_\theta $ |
$ S $ |
$ \theta $ |
$ \mathbb{A}^{ {\mathbb{Z}}} $ |
$ X_\theta $ |
$ \boldsymbol{u}: {\mathbb{N}}\to {\mathbb{C}} $ |
$ \lim\limits_{N\to\infty}\frac1N\sum\limits_{n\leq N}f(S^nx) \boldsymbol{u}(n) = 0 $ |
$ f\in C(X_\theta) $ |
$ x\in X_\theta $ |
References:
[1] |
H. El Abdalaoui, J. Kułaga-Przymus, M. 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. |
[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. |
[3] |
J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511546563. |
[4] |
V. Bergelson, J. Kułaga-Przymus, M. 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. |
[5] |
V. Bergelson, J. Kułaga-Przymus, M. 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. |
[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. |
[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. |
[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. |
[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. |
[10] |
J.-M. Deshouillers, M. 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. |
[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. |
[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. |
[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. |
[14] |
N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b13861. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
J. L. Herning, Spectrum and Factors of Substitution Dynamical Systems, Ph.D dissertation, George Washington University, 2013, 93 pp. |
[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. |
[23] |
I. Kátai,
A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.
doi: 10.1007/BF01949145. |
[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. |
[25] |
J. Konieczny, Möbius orthogonality for $q$-multiplicative sequences, preprint, arXiv: 1808.06196. |
[26] |
J. Konieczny,
On multiplicative automatic sequences, Bulletin of the London Mathematical Society, 52 (2020), 175-184.
doi: 10.1112/blms.12317. |
[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. |
[28] |
E. Lehrer,
Toplogical mixing and uniquely ergodic models, Israel J. Math., 57 (1987), 239-255.
doi: 10.1007/BF02772176. |
[29] |
M. Lemańczyk and M. K. Mentzen,
On metric properties of substitutions, Compositio Math., 65 (1988), 241-263.
|
[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. |
[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. |
[32] |
K. Matomäki, M. 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. |
[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. |
[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. |
[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. |
[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. |
[37] |
C. Müllner,
Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219-3290.
doi: 10.1215/00127094-2017-0024. |
[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. |
[39] |
P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sites/default/files/Mobius%20lectures%20Summer%202010.pdf. |
[40] |
J.-C. Schlage-Puchta, Completely multiplicative automatic functions, Integers, 11 (2011), A31, 8 pp.
doi: 10.1515/INTEG.2011.055. |
[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/ |
[42] |
W. A. Veech,
A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341.
doi: 10.1007/BF01667386. |
[43] |
S. Yazdani,
Multiplicative functions and k-automatic sequences, J. Théor. Nombres Bordeaux, 13 (2001), 651-658.
doi: 10.5802/jtnb.342. |
show all references
References:
[1] |
H. El Abdalaoui, J. Kułaga-Przymus, M. 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. |
[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. |
[3] |
J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511546563. |
[4] |
V. Bergelson, J. Kułaga-Przymus, M. 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. |
[5] |
V. Bergelson, J. Kułaga-Przymus, M. 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. |
[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. |
[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. |
[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. |
[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. |
[10] |
J.-M. Deshouillers, M. 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. |
[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. |
[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. |
[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. |
[14] |
N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b13861. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[21] |
J. L. Herning, Spectrum and Factors of Substitution Dynamical Systems, Ph.D dissertation, George Washington University, 2013, 93 pp. |
[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. |
[23] |
I. Kátai,
A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.
doi: 10.1007/BF01949145. |
[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. |
[25] |
J. Konieczny, Möbius orthogonality for $q$-multiplicative sequences, preprint, arXiv: 1808.06196. |
[26] |
J. Konieczny,
On multiplicative automatic sequences, Bulletin of the London Mathematical Society, 52 (2020), 175-184.
doi: 10.1112/blms.12317. |
[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. |
[28] |
E. Lehrer,
Toplogical mixing and uniquely ergodic models, Israel J. Math., 57 (1987), 239-255.
doi: 10.1007/BF02772176. |
[29] |
M. Lemańczyk and M. K. Mentzen,
On metric properties of substitutions, Compositio Math., 65 (1988), 241-263.
|
[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. |
[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. |
[32] |
K. Matomäki, M. 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. |
[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. |
[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. |
[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. |
[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. |
[37] |
C. Müllner,
Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219-3290.
doi: 10.1215/00127094-2017-0024. |
[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. |
[39] |
P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sites/default/files/Mobius%20lectures%20Summer%202010.pdf. |
[40] |
J.-C. Schlage-Puchta, Completely multiplicative automatic functions, Integers, 11 (2011), A31, 8 pp.
doi: 10.1515/INTEG.2011.055. |
[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/ |
[42] |
W. A. Veech,
A criterion for a process to be prime, Monatsh. Math., 94 (1982), 335-341.
doi: 10.1007/BF01667386. |
[43] |
S. Yazdani,
Multiplicative functions and k-automatic sequences, J. Théor. Nombres Bordeaux, 13 (2001), 651-658.
doi: 10.5802/jtnb.342. |
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