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

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