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.

[2]

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

[3]

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

[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.

[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.

[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.

[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. 

[8]

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

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[19]

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

[20]

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

[21]

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

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[30]

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

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.

[2]

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

[3]

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

[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.

[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.

[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.

[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. 

[8]

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

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[19]

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

[20]

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

[21]

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

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[30]

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

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