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June  2017, 37(6): 2899-2944. doi: 10.3934/dcds.2017125

The Chowla and the Sarnak conjectures from ergodic theory point of view

El Houcein El Abdalaoui 1, , Joanna Kułaga-Przymus 2, , Mariusz Lemańczyk 3, and  Thierry de la Rue 4,

1. 

Laboratoire de Mathématiques Raphaël Salem, Normandie Université, Université de Rouen, CNRS, Avenue de l'Université, 76801 Saint Etienne du Rouvray, France
2. 

Institute of Mathematics, Polish Acadamy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland, and, Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
3. 

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
4. 

Laboratoire de Mathématiques Raphaël Salem, Normandie Université, Université de Rouen, CNRS, Avenue de l'Université, 76801 Saint Etienne du Rouvray, France

Received  April 2015 Revised  January 2017 Published  February 2017

We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences in $\{-1,0,1\}^{{\mathbb{N}^*}}$, and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic theory.

Keywords: Chowla conjecture, Sarnak conjecture, Möbius orthogonality, ergodic theory, theory of joinings.
Mathematics Subject Classification: Primary: 37A45, 37B10; Secondary: 11N37.
Citation: El Houcein El Abdalaoui, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Thierry de la Rue. The Chowla and the Sarnak conjectures from ergodic theory point of view. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2899-2944. doi: 10.3934/dcds.2017125
References:
[1]

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.  doi: 10.1090/proc/12683.

[2]

H. El Abdalaoui, J. Kulaga-Przymus, M. Lemańczyk and T. de la Rue, The howla and the Sarnak conjectures from ergodic theory point of view (extended version), arXiv: 1410.1673.

[3]

H. El Abdalaoui, M. Lemańczyk and T. de la Rue, A dynamical point of view on the set of $ \mathscr{B}$-free integers, International Mathematics Research Notices, 2015 (2015), 7258-7286.  doi: 10.1093/imrn/rnu164.

[4]

T. M. Apostol, Introduction to Analytic Number Theory Springer-Verlag, New York-Heidelberg, 1976, Undergraduate Texts in Mathematics.

[5]

F. Cellarosi and Y. G. Sinai, Ergodic properties of square-free numbers, J. Eur. Math. Soc., 15 (2013), 1343-1374.  doi: 10.4171/JEMS/394.

[6]

S. Chowla, The Riemann Hypothesis and Hilbert's Tenth Problem Mathematics and Its Applications, Vol. 4, Gordon and Breach Science Publishers, New York, 1965.

[7]

H. Davenport, On some infinite series involving arithmetical functions. Ⅱ, Quart. J. Math. Oxford, 8 (1937), 313-320.  doi: 10.1093/qmath/os-8.1.313.

[8]

T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math., 74 (1991), 241-256.  doi: 10.1007/BF02775789.

[9]

T. Downarowicz, Survey of odometers and Toeplitz flows, in Algebraic and Topological Dynamics, vol. 385 of Contemp. Math. , Amer. Math. Soc. , Providence, RI, 2005, 7-37. doi: 10.1090/conm/385/07188.

[10]

T. Downarowicz, Entropy in Dynamical Systems vol. 18 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155.

[11]

T. Downarowicz and S. Kasjan, Odometers and Toeplitz subshifts revisited in the context of Sarnak's conjecture, to appear in Studia Math., 229 (2015), 45-72, arXiv: 1502.02307.

[12]

N. P. Fogg, Substitutions in ynamics, rithmetics and ombinatorics vol. 1794 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002, Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. doi: 10.1007/b13861.

[13]

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

[14]

K. Jacobs and M. Keane, 0-1-sequences of Toeplitz type, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 13 (1969), 123-131.  doi: 10.1007/BF00537017.

[15]

T. Kamae, Subsequences of normal sequences, Israel J. Math., 16 (1973), 121-149.  doi: 10.1007/BF02757864.

[16]

D. Kerr and H. Li, Independence in topological and C*-dynamics, Math. Ann., 338 (2007), 869-926.  doi: 10.1007/s00208-007-0097-z.

[17]

D. Kwietniak, Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts, Discrete Contin. Dyn. Syst., 33 (2013), 2451-2467.  doi: 10.3934/dcds.2013.33.2451.

[18]

K. Matomäki, M. Radziwiłł and T. Tao, Sign patterns of the iouville and Möbius functions Forum of Mathematics, Sigma 4 e14, (2016), 44 pp. doi: 10.1017/fms.2016.6.

[19]

L. Mirsky, Arithmetical pattern problems relating to divisibility by r th powers, Proc. London Math. Soc.(2), 50 (1949), 497-508.  doi: 10.1112/plms/s2-50.7.497.

[20]

D. Ornstein, Factors of ernoulli shifts are ernoulli shifts, Advances in Math., 5 (1970), 349-364 (1970).  doi: 10.1016/0001-8708(70)90009-5.

[21]

W. Parry, Entropy and Generators in Ergodic Theory W. A. Benjamin, Inc. , New York-Amsterdam, 1969.

[22]

R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow, to appear in Israel J. Math., 210 (2015), 335-357, arXiv: 1205.2905. doi: 10.1007/s11856-015-1255-8.

[23]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sarnak/.

[24]

P. C. Shields, The Ergodic Theory of Discrete Sample Paths vol. 13 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1996. doi: 10.1090/gsm/013.

[25]

T. Tao, The howla conjecture and the Sarnak conjecture, What's new (blog) http://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-the-sarnak-conjecture.

[26]

J.-P. Thouvenot, Une classe de systémes pour lesquels la conjecture de insker est vraie, Israel J. Math., 21 (1975), 208-214, Conference on Ergodic Theory and Topological Dynamics (Kibbutz Lavi, 1974).  doi: 10.1007/BF02760798.

[27]

E. C. Titchmarsh, The Theory of the iemann Zeta-function 2nd edition, The Clarendon Press Oxford University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown.

[28]

P. Walters, An Introduction to Ergodic Theory vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982.

[29]

B. Weiss, Normal sequences as collectives, in Proc. Symp. on Topological Dynamics and ergodic theory Univ. of Kentucky, 1971.

[30]

B. Weiss, Single Orbit Dynamics vol. 95 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 2000.

[31]

S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete, 67 (1984), 95-107.  doi: 10.1007/BF00534085.

show all references

References:
[1]

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.  doi: 10.1090/proc/12683.

[2]

H. El Abdalaoui, J. Kulaga-Przymus, M. Lemańczyk and T. de la Rue, The howla and the Sarnak conjectures from ergodic theory point of view (extended version), arXiv: 1410.1673.

[3]

H. El Abdalaoui, M. Lemańczyk and T. de la Rue, A dynamical point of view on the set of $ \mathscr{B}$-free integers, International Mathematics Research Notices, 2015 (2015), 7258-7286.  doi: 10.1093/imrn/rnu164.

[4]

T. M. Apostol, Introduction to Analytic Number Theory Springer-Verlag, New York-Heidelberg, 1976, Undergraduate Texts in Mathematics.

[5]

F. Cellarosi and Y. G. Sinai, Ergodic properties of square-free numbers, J. Eur. Math. Soc., 15 (2013), 1343-1374.  doi: 10.4171/JEMS/394.

[6]

S. Chowla, The Riemann Hypothesis and Hilbert's Tenth Problem Mathematics and Its Applications, Vol. 4, Gordon and Breach Science Publishers, New York, 1965.

[7]

H. Davenport, On some infinite series involving arithmetical functions. Ⅱ, Quart. J. Math. Oxford, 8 (1937), 313-320.  doi: 10.1093/qmath/os-8.1.313.

[8]

T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math., 74 (1991), 241-256.  doi: 10.1007/BF02775789.

[9]

T. Downarowicz, Survey of odometers and Toeplitz flows, in Algebraic and Topological Dynamics, vol. 385 of Contemp. Math. , Amer. Math. Soc. , Providence, RI, 2005, 7-37. doi: 10.1090/conm/385/07188.

[10]

T. Downarowicz, Entropy in Dynamical Systems vol. 18 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2011. doi: 10.1017/CBO9780511976155.

[11]

T. Downarowicz and S. Kasjan, Odometers and Toeplitz subshifts revisited in the context of Sarnak's conjecture, to appear in Studia Math., 229 (2015), 45-72, arXiv: 1502.02307.

[12]

N. P. Fogg, Substitutions in ynamics, rithmetics and ombinatorics vol. 1794 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002, Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. doi: 10.1007/b13861.

[13]

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

[14]

K. Jacobs and M. Keane, 0-1-sequences of Toeplitz type, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 13 (1969), 123-131.  doi: 10.1007/BF00537017.

[15]

T. Kamae, Subsequences of normal sequences, Israel J. Math., 16 (1973), 121-149.  doi: 10.1007/BF02757864.

[16]

D. Kerr and H. Li, Independence in topological and C*-dynamics, Math. Ann., 338 (2007), 869-926.  doi: 10.1007/s00208-007-0097-z.

[17]

D. Kwietniak, Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts, Discrete Contin. Dyn. Syst., 33 (2013), 2451-2467.  doi: 10.3934/dcds.2013.33.2451.

[18]

K. Matomäki, M. Radziwiłł and T. Tao, Sign patterns of the iouville and Möbius functions Forum of Mathematics, Sigma 4 e14, (2016), 44 pp. doi: 10.1017/fms.2016.6.

[19]

L. Mirsky, Arithmetical pattern problems relating to divisibility by r th powers, Proc. London Math. Soc.(2), 50 (1949), 497-508.  doi: 10.1112/plms/s2-50.7.497.

[20]

D. Ornstein, Factors of ernoulli shifts are ernoulli shifts, Advances in Math., 5 (1970), 349-364 (1970).  doi: 10.1016/0001-8708(70)90009-5.

[21]

W. Parry, Entropy and Generators in Ergodic Theory W. A. Benjamin, Inc. , New York-Amsterdam, 1969.

[22]

R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow, to appear in Israel J. Math., 210 (2015), 335-357, arXiv: 1205.2905. doi: 10.1007/s11856-015-1255-8.

[23]

P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sarnak/.

[24]

P. C. Shields, The Ergodic Theory of Discrete Sample Paths vol. 13 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1996. doi: 10.1090/gsm/013.

[25]

T. Tao, The howla conjecture and the Sarnak conjecture, What's new (blog) http://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-the-sarnak-conjecture.

[26]

J.-P. Thouvenot, Une classe de systémes pour lesquels la conjecture de insker est vraie, Israel J. Math., 21 (1975), 208-214, Conference on Ergodic Theory and Topological Dynamics (Kibbutz Lavi, 1974).  doi: 10.1007/BF02760798.

[27]

E. C. Titchmarsh, The Theory of the iemann Zeta-function 2nd edition, The Clarendon Press Oxford University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown.

[28]

P. Walters, An Introduction to Ergodic Theory vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1982.

[29]

B. Weiss, Normal sequences as collectives, in Proc. Symp. on Topological Dynamics and ergodic theory Univ. of Kentucky, 1971.

[30]

B. Weiss, Single Orbit Dynamics vol. 95 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 2000.

[31]

S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete, 67 (1984), 95-107.  doi: 10.1007/BF00534085.

Figure 1.  Sturmian sequence
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