2021, 17: 529-555. doi: 10.3934/jmd.2021018

On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao

1. 

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

2. 

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

Received  August 25, 2020 Revised  August 24, 2021 Published  November 2021

It is shown that in a class of counterexamples to Elliott's conjecture by Matomäki, Radziwiłł, and Tao [23] the Chowla conjecture holds along a subsequence.

Citation: Alexander Gomilko, Mariusz Lemańczyk, Thierry de la Rue. On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao. Journal of Modern Dynamics, 2021, 17: 529-555. doi: 10.3934/jmd.2021018
References:
[1]

H. El Abdalaoui, M. Lemańczyk and T. de la Rue, A dynamical point of view on the set of $\mathcal{B}$-free integers, Int. Math. Res. Not. IMRN, (2015), 7258–7286. doi: 10.1093/imrn/rnu164.  Google Scholar

[2]

L. M. Abramov, Metric automorphisms with quasi-discrete spectrum, Izv. Akad. Nauk SSSR Ser. Mat., 26 (1962), 513-530.   Google Scholar

[3]

American Institute of Mathematics, workshop, Sarnak's Conjecture, December 2018. Available from: http://aimpl.org/sarnakconjecture/3/. Google Scholar

[4]

V. BergelsonJ. Kulaga-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]

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

[6]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[7]

P. D. T. A. Elliott, Multiplicative functions $|g| \leq 1$ and their convolutions: An overview, Séminaire de Théorie des Nombres, Paris 1987-88. Progr. Math., 81 (1990), 63-75.   Google Scholar

[8]

P. D. T. A. Elliott, On the correlation of multiplicative functions, Notas Soc. Mat. Chile, 11 (1992), 1-11.   Google Scholar

[9]

P. D. T. A. Elliott, On the correlation of multiplicative and the sum of additive arithmetic functions, Mem. Amer. Math. Soc., 112 (1994). doi: 10.1090/memo/0538.  Google Scholar

[10]

L. Flaminio, Mixing k-fold independent processes of zero entropy, Proc. Amer. Math. Soc., 118 (1993), 1263-1269.  doi: 10.2307/2160087.  Google Scholar

[11]

N. Frantzikinakis, Ergodicity of the Liouville system implies the Chowla conjecture, Discrete Anal., (2017), 19, 41 pp. doi: 10.19086/da.2733.  Google Scholar

[12]

N. Frantzikinakis, An averaged Chowla and Elliott conjecture along independent polynomials, Int. Math. Res. Not. IMRN, (2018), 3721–3743. doi: 10.1093/imrn/rnx002.  Google Scholar

[13]

N. Frantzikinkis and B. Host, Asymptotics for multilinear averages of multiplicative functions, Math. Proc. Cambridge Philos. Soc., 161 (2016), 87-101.  doi: 10.1017/S0305004116000116.  Google Scholar

[14]

N. Frantzikinakis and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math. (2), 187 (2018), 869-931.  doi: 10.4007/annals.2018.187.3.6.  Google Scholar

[15]

N. Frantzikinakis and B. Host, Furstenberg systems of bounded multiplicative functions and applications, Int. Math. Res. Not. IMRN, (2021), 6077–6107. doi: 10.1093/imrn/rnz037.  Google Scholar

[16]

H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601.  doi: 10.2307/2372899.  Google Scholar

[17]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[18]

F. Hahn and W. Parry, Minimal dynamical systems with quasi-discrete spectrum, J. London Math. Soc., 40 (1965), 309-323.  doi: 10.1112/jlms/s1-40.1.309.  Google Scholar

[19]

E. Jenvey, Strong stationarity and de Finetti's theorem, J. Anal. Math., 73 (1997), 1-18.  doi: 10.1007/BF02788136.  Google Scholar

[20]

O. Klurman, Correlations of multiplicative functions and applications, Compo. Math., 153 (2017), 1622-1657.  doi: 10.1112/S0010437X17007163.  Google Scholar

[21]

L. Matthiesen, Linear correlations of multiplicative functions, Proc. Lond. Math. Soc., 121 (2020), 372-425.  doi: 10.1112/plms.12309.  Google Scholar

[22]

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

[23]

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

[24]

J. Rivat, Bases of Analytic Number Theory, Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, 1–113, Lecture Notes in Math., 2213, Springer, Cham, 2018., doi: 10.1007/978-3-319-74908-2.  Google Scholar

[25]

P. Sarnak, Three Lectures on the Möbius Function, Randomness and Dynamics., Available from: http://publications.ias.edu/sarnak/. Google Scholar

[26]

T. Tao, The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, Forum Math. Pi, 4 (2016), 36 pp. doi: 10.1017/fmp.2016.6.  Google Scholar

[27]

T. Tao and J. Teräväinen, Odd order cases of the logarithmically averaged Chowla conjecture, J. Théor. Nombres Bordeaux, 30 (2018), 997-1015.  doi: 10.5802/jtnb.1062.  Google Scholar

[28]

T. Tao and J. Teräväinen, The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures, Duke Math. J., 168 (2019), 1977-2027.  doi: 10.1215/00127094-2019-0002.  Google Scholar

[29]

T. Tao and J. Teräväinen, The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures, Algebra Number Theory, 13 (2019), 2103-2150.  doi: 10.2140/ant.2019.13.2103.  Google Scholar

[30]

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

show all references

References:
[1]

H. El Abdalaoui, M. Lemańczyk and T. de la Rue, A dynamical point of view on the set of $\mathcal{B}$-free integers, Int. Math. Res. Not. IMRN, (2015), 7258–7286. doi: 10.1093/imrn/rnu164.  Google Scholar

[2]

L. M. Abramov, Metric automorphisms with quasi-discrete spectrum, Izv. Akad. Nauk SSSR Ser. Mat., 26 (1962), 513-530.   Google Scholar

[3]

American Institute of Mathematics, workshop, Sarnak's Conjecture, December 2018. Available from: http://aimpl.org/sarnakconjecture/3/. Google Scholar

[4]

V. BergelsonJ. Kulaga-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]

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

[6]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[7]

P. D. T. A. Elliott, Multiplicative functions $|g| \leq 1$ and their convolutions: An overview, Séminaire de Théorie des Nombres, Paris 1987-88. Progr. Math., 81 (1990), 63-75.   Google Scholar

[8]

P. D. T. A. Elliott, On the correlation of multiplicative functions, Notas Soc. Mat. Chile, 11 (1992), 1-11.   Google Scholar

[9]

P. D. T. A. Elliott, On the correlation of multiplicative and the sum of additive arithmetic functions, Mem. Amer. Math. Soc., 112 (1994). doi: 10.1090/memo/0538.  Google Scholar

[10]

L. Flaminio, Mixing k-fold independent processes of zero entropy, Proc. Amer. Math. Soc., 118 (1993), 1263-1269.  doi: 10.2307/2160087.  Google Scholar

[11]

N. Frantzikinakis, Ergodicity of the Liouville system implies the Chowla conjecture, Discrete Anal., (2017), 19, 41 pp. doi: 10.19086/da.2733.  Google Scholar

[12]

N. Frantzikinakis, An averaged Chowla and Elliott conjecture along independent polynomials, Int. Math. Res. Not. IMRN, (2018), 3721–3743. doi: 10.1093/imrn/rnx002.  Google Scholar

[13]

N. Frantzikinkis and B. Host, Asymptotics for multilinear averages of multiplicative functions, Math. Proc. Cambridge Philos. Soc., 161 (2016), 87-101.  doi: 10.1017/S0305004116000116.  Google Scholar

[14]

N. Frantzikinakis and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math. (2), 187 (2018), 869-931.  doi: 10.4007/annals.2018.187.3.6.  Google Scholar

[15]

N. Frantzikinakis and B. Host, Furstenberg systems of bounded multiplicative functions and applications, Int. Math. Res. Not. IMRN, (2021), 6077–6107. doi: 10.1093/imrn/rnz037.  Google Scholar

[16]

H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math., 83 (1961), 573-601.  doi: 10.2307/2372899.  Google Scholar

[17]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[18]

F. Hahn and W. Parry, Minimal dynamical systems with quasi-discrete spectrum, J. London Math. Soc., 40 (1965), 309-323.  doi: 10.1112/jlms/s1-40.1.309.  Google Scholar

[19]

E. Jenvey, Strong stationarity and de Finetti's theorem, J. Anal. Math., 73 (1997), 1-18.  doi: 10.1007/BF02788136.  Google Scholar

[20]

O. Klurman, Correlations of multiplicative functions and applications, Compo. Math., 153 (2017), 1622-1657.  doi: 10.1112/S0010437X17007163.  Google Scholar

[21]

L. Matthiesen, Linear correlations of multiplicative functions, Proc. Lond. Math. Soc., 121 (2020), 372-425.  doi: 10.1112/plms.12309.  Google Scholar

[22]

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

[23]

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

[24]

J. Rivat, Bases of Analytic Number Theory, Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, 1–113, Lecture Notes in Math., 2213, Springer, Cham, 2018., doi: 10.1007/978-3-319-74908-2.  Google Scholar

[25]

P. Sarnak, Three Lectures on the Möbius Function, Randomness and Dynamics., Available from: http://publications.ias.edu/sarnak/. Google Scholar

[26]

T. Tao, The logarithmically averaged Chowla and Elliott conjectures for two-point correlations, Forum Math. Pi, 4 (2016), 36 pp. doi: 10.1017/fmp.2016.6.  Google Scholar

[27]

T. Tao and J. Teräväinen, Odd order cases of the logarithmically averaged Chowla conjecture, J. Théor. Nombres Bordeaux, 30 (2018), 997-1015.  doi: 10.5802/jtnb.1062.  Google Scholar

[28]

T. Tao and J. Teräväinen, The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures, Duke Math. J., 168 (2019), 1977-2027.  doi: 10.1215/00127094-2019-0002.  Google Scholar

[29]

T. Tao and J. Teräväinen, The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures, Algebra Number Theory, 13 (2019), 2103-2150.  doi: 10.2140/ant.2019.13.2103.  Google Scholar

[30]

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

[1]

Nikolai Edeko. On the isomorphism problem for non-minimal transformations with discrete spectrum. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 6001-6021. doi: 10.3934/dcds.2019262

[2]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795

[3]

Mariusz Lemańczyk, Clemens Müllner. Automatic sequences are orthogonal to aperiodic multiplicative functions. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6877-6918. doi: 10.3934/dcds.2020260

[4]

Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475

[5]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098

[6]

Prof. Dr.rer.nat Widodo. Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 191-208. doi: 10.3934/dcds.2002.8.191

[7]

Nguyen Huy Chieu, Jen-Chih Yao. Subgradients of the optimal value function in a parametric discrete optimal control problem. Journal of Industrial & Management Optimization, 2010, 6 (2) : 401-410. doi: 10.3934/jimo.2010.6.401

[8]

Giovanni Colombo, Thuy T. T. Le. Higher order discrete controllability and the approximation of the minimum time function. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4293-4322. doi: 10.3934/dcds.2015.35.4293

[9]

Feng Ma, Jiansheng Shu, Yaxiong Li, Jian Wu. The dual step size of the alternating direction method can be larger than 1.618 when one function is strongly convex. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1173-1185. doi: 10.3934/jimo.2020016

[10]

Andi Kivinukk, Anna Saksa. On Rogosinski-type approximation processes in Banach space using the framework of the cosine operator function. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021030

[11]

Qi Wang. Global solutions of a Keller--Segel system with saturated logarithmic sensitivity function. Communications on Pure & Applied Analysis, 2015, 14 (2) : 383-396. doi: 10.3934/cpaa.2015.14.383

[12]

Jiyoung Han. Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2205-2225. doi: 10.3934/dcds.2020359

[13]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939

[14]

Xilin Fu, Zhang Chen. New discrete analogue of neural networks with nonlinear amplification function and its periodic dynamic analysis. Conference Publications, 2007, 2007 (Special) : 391-398. doi: 10.3934/proc.2007.2007.391

[15]

Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020119

[16]

J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413

[17]

H. N. Mhaskar, T. Poggio. Function approximation by deep networks. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4085-4095. doi: 10.3934/cpaa.2020181

[18]

Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669

[19]

Ken Ono. Parity of the partition function. Electronic Research Announcements, 1995, 1: 35-42.

[20]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741

2020 Impact Factor: 0.848

Metrics

  • PDF downloads (30)
  • HTML views (18)
  • Cited by (0)

[Back to Top]