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August  2018, 38(8): 4087-4115. doi: 10.3934/dcds.2018178

Automatic sequences as good weights for ergodic theorems

1. 

Institute of Mathematics, University of Leipzig, P.O. Box 100 920, 04009 Leipzig, Germany

2. 

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel

3. 

Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Łojasiewicza 6, 30-348 Kraków, Poland

Received  November 2017 Revised  March 2018 Published  May 2018

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems. We show that automatic sequences are good weights in $ L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $ L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $ L^r$, $ r>1$.

Citation: Tanja Eisner, Jakub Konieczny. Automatic sequences as good weights for ergodic theorems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4087-4115. doi: 10.3934/dcds.2018178
References:
[1]

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoret. Comput. Sci., 98 (1992), 163–197. doi: 10.1016/0304-3975(92)90001-V. Google Scholar

[2]

J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563. Google Scholar

[3]

I. Assani, A weighted pointwise ergodic theorem, Ann. Inst. H. Poincaré Probab. Statist., 34 (1998), 139–150. doi: 10.1016/S0246-0203(98)80021-6. Google Scholar

[4]

I. Assani, Wiener Wintner Ergodic Theorems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/4538. Google Scholar

[5]

I. Assani and R. Moore, A good universal weight for multiple recurrence averages with commuting transformations in norm, Ergodic Theory Dynam. Systems, 37 (2017), 1009–1025, available at https://arXiv.org/abs/1506.06730. doi: 10.1017/etds.2015.76. Google Scholar

[6]

I. Assani and K. Presser, A survey of the return times theorem, in Ergodic Theory and Dynamical Systems, De Gruyter Proc. Math., De Gruyter, Berlin, 2014, 19–58. Google Scholar

[7]

J. P. Bell, M. Coons and K. G. Hare, The minimal growth of a k-regular sequence, Bull. Aust. Math. Soc., 90 (2014), 195–203. doi: 10.1017/S0004972714000197. Google Scholar

[8]

J. P. Bell, M. Coons and K. G. Hare, Growth degree classification for finitely generated semigroups of integer matrices, Semigroup Forum, 92 (2016), 23–44. doi: 10.1007/s00233-015-9725-1. Google Scholar

[9]

A. Bellow and V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, Trans. Amer. Math. Soc., 288 (1985), 307–345, URL http://dx.doi.org/10.2307/2000442. doi: 10.1090/S0002-9947-1985-0773063-8. Google Scholar

[10]

D. Berend, M. Lin, J. Rosenblatt and A. Tempelman, Modulated and subsequential ergodic theorems in Hilbert and Banach spaces, Ergodic Theory Dynam. Systems, 22 (2002), 1653–1665. doi: 10.1017/S0143385702000846. Google Scholar

[11]

J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Publ. Math., Inst. Hautes Études Sci., 69 (1985), 5–45, URL http://www.numdam.org/item?id=PMIHES_1989__69__5_0, With an appendix by the author, H. Furstenberg, Y. Katznelson and D. S. Ornstein. Google Scholar

[12]

J. Bourgain, H. Furstenberg, Y. Katznelson and D. S. Ornstein, Appendix on return-time sequences, Publ. Math., Inst. Hautes Études Sci., 69 (1985), 42–45, URL http://www.numdam.org/item?id=PMIHES_1989__69__42_0. Google Scholar

[13]

Z. Buczolich and R. D. Mauldin, Divergent square averages, Ann. of Math. (2), 171 (2010), 1479–1530. doi: 10.4007/annals.2010.171.1479. Google Scholar

[14]

Q. Chu, Convergence of weighted polynomial multiple ergodic averages, Proc. Amer. Math. Soc., 137 (2009), 1363–1369. doi: 10.1090/S0002-9939-08-09614-7. Google Scholar

[15]

D. Cömez, M. Lin and J. Olsen, Weighted ergodic theorems for mean ergodic L1- contractions, Trans. Amer. Math. Soc., 350 (1998), 101–117. doi: 10.1090/S0002-9947-98-01986-2. Google Scholar

[16]

C. Cuny and M. Weber, Ergodic theorems with arithmetical weights, Israel J. Math., 217 (2017), 139-180. doi: 10.1007/s11856-017-1441-y. Google Scholar

[17]

S. Drappeau and C. Müllner, Exponential sums with automatic sequences, 2017, Preprint, available at https://arXiv.org/abs/1710.01091.Google Scholar

[18]

M. Drmota and J. F. Morgenbesser, Generalized Thue-Morse sequences of squares, Israel J. Math., 190 (2012), 157–193. doi: 10.1007/s11856-011-0186-2. Google Scholar

[19]

P. Dumas, Joint spectral radius, dilation equations, and asymptotic behavior of radix-rational sequences, Linear Algebra Appl., 438 (2013), 2107–2126. doi: 10.1016/j.laa.2012.10.013. Google Scholar

[20]

P. Dumas, Asymptotic expansions for linear homogeneous divide-and-conquer recurrences: algebraic and analytic approaches collated, Theoret. Comput. Sci., 548 (2014), 25–53. doi: 10.1016/j.tcs.2014.06.036. Google Scholar

[21]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, vol. 259 of Graduate Texts in Mathematics, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2. Google Scholar

[22]

T. Eisner, A polynomial version of Sarnak's conjecture, C. R. Math. Acad. Sci. Paris, 353 (2015), 569–572. doi: 10.1016/j.crma.2015.04.009. Google Scholar

[23]

T. Eisner, Linear sequences and weighted ergodic theorems, Abstr. Appl. Anal., (2013), Art. ID 815726, 5 pp. Google Scholar

[24]

T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, vol. 272 of Graduate Texts in Mathematics, Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2. Google Scholar

[25]

T. Eisner and B. Krause, (Uniform) convergence of twisted ergodic averages, Ergodic Theory Dynam. Systems, 36 (2016), 2172–2202. doi: 10.1017/etds.2015.6. Google Scholar

[26]

T. Eisner and P. Zorin-Kranich, Uniformity in the Wiener-Wintner theorem for nilsequences, Discrete Contin. Dyn. Syst., 33 (2013), 3497–3516. doi: 10.3934/dcds.2013.33.3497. Google Scholar

[27]

E. H. El Abdalaoui, J. Ku laga-Przymus, M. Lemańczyk and T. de la Rue, The Chowla and the Sarnak conjectures from ergodic theory point of view, Discrete Contin. Dyn. Syst., 37 (2017), 2899–2944. doi: 10.3934/dcds.2017125. Google Scholar

[28]

A.-H. Fan, Weighted Birkhoff ergodic theorem with oscillating weights, Ergodic Theory and Dynamical Systems, (2017), 1-15. doi: 10.1017/etds.2017.81. Google Scholar

[29]

N. Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061–1071. doi: 10.1017/S0143385706000204. Google Scholar

[30]

A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., 13 (1967/1968), 259-265. doi: 10.4064/aa-13-3-259-265. Google Scholar

[31]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2), 175 (2012), 465–540. doi: 10.4007/annals.2012.175.2.2. Google Scholar

[32]

B. Host and B. Kra, Uniformity seminorms on $ \ell^∞$ and applications, J. Anal. Math., 108 (2009), 219–276. doi: 10.1007/s11854-009-0024-1. Google Scholar

[33]

J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, 2017, Preprint, available at https://arXiv.org/abs/1611.09985.Google Scholar

[34]

B. Krause and P. Zorin-Kranich, A random pointwise ergodic theorem with Hardy field weights, Illinois J. Math., 59 (2015), 663–674, URL http://projecteuclid.org/euclid.ijm/1475266402. Google Scholar

[35]

P. LaVictoire, Universally L1-bad arithmetic sequences, J. Anal. Math., 113 (2011), 241–263. doi: 10.1007/s11854-011-0006-y. Google Scholar

[36]

E. Lesigne, Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener-Wintner, Ergodic Theory Dynam. Systems, 10 (1990), 513– 521. doi: 10.1017/S014338570000571X. Google Scholar

[37]

E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784. Google Scholar

[38]

E. Lesigne and C. Mauduit, Propriétés ergodiques des suites q-multiplicatives, Compositio Math., 100 (1996), 131–169, URL http://www.numdam.org/item?id=CM_1996__100_2_131_0. Google Scholar

[39]

E. Lesigne, C. Mauduit and B. Mossé, Le théorème ergodique le long d'une suite q-multiplicative, Compositio Math., 93 (1994), 49–79, URL http://www.numdam.org/item?id=CM_1994__93_1_49_0. Google Scholar

[40]

M. Lin, J. Olsen and A. Tempelman, On modulated ergodic theorems for Dunford-Schwartz operators, in Proceedings of the Conference on Probability, Ergodic Theory, and Analysis (Evanston, IL, 1997), 43 (1999), 542–567, URL http://projecteuclid.org/euclid.ijm/1255985110. Google Scholar

[41]

B. Martin, C. Mauduit and J. Rivat, Théorème des nombres premiers pour les fonctions digitales, Acta Arith., 165 (2014), 11–45. doi: 10.4064/aa165-1-2. Google Scholar

[42]

C. Mauduit, Automates finis et ensembles normaux, Ann. Inst. Fourier (Grenoble), 36 (1986), 1–25, URL http://www.numdam.org/item?id=AIF_1986__36_2_1_0. doi: 10.5802/aif.1044. Google Scholar

[43]

C. Mauduit, Propriétés arithmétiques des substitutions et automates infinis, Ann. Inst. Fourier (Grenoble), 56 (2006), 2525–2549, URL http://aif.cedram.org/item?id=AIF_200__56_7_2525_0, Numération, pavages, substitutions. doi: 10.5802/aif.2248. Google Scholar

[44]

C. Mauduit and A. Sárközy, On finite pseudorandom binary sequences. Ⅱ. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences, a further construction, J. Number Theory, 73 (1998), 256–276. doi: 10.1006/jnth.1998.2286. Google Scholar

[45]

C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219–3290. doi: 10.1215/00127094-2017-0024. Google Scholar

[46]

C. Müllner, Exponential Sum Estimates and Fourier Analytic Methods for Digitally Based Dynamical Systems, PhD thesis, Technische Universit at Wien, 2017.Google Scholar

[47]

K. Petersen, Ergodic Theory, vol. 2 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1989, Corrected reprint of the 1983 original. Google Scholar

[48]

M. Queffélec, Substitution Dynamical Systems—Spectral Analysis, vol. 1294 of Lecture Notes in Mathematics, 2nd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6. Google Scholar

[49]

T. Tao, Poincaré's Legacies, Pages from Year two of a Mathematical blog. Part I, American Mathematical Society, Providence, RI, 2009. Google Scholar

[50]

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

[51]

N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415–426. doi: 10.2307/2371534. Google Scholar

[52]

M. Wierdl, Pointwise ergodic theorems along the prime numbers, Israel J. Math., 64 (1988), 315-336. doi: 10.1007/BF02882425. Google Scholar

[53]

P. Zorin-Kranich, A double return times theorem, Israel J. Math., 204 (2014), 85–96, available at https://arXiv.org/abs/1506.05748. doi: 10.1007/s11856-014-1112-1. Google Scholar

show all references

References:
[1]

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoret. Comput. Sci., 98 (1992), 163–197. doi: 10.1016/0304-3975(92)90001-V. Google Scholar

[2]

J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563. Google Scholar

[3]

I. Assani, A weighted pointwise ergodic theorem, Ann. Inst. H. Poincaré Probab. Statist., 34 (1998), 139–150. doi: 10.1016/S0246-0203(98)80021-6. Google Scholar

[4]

I. Assani, Wiener Wintner Ergodic Theorems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/4538. Google Scholar

[5]

I. Assani and R. Moore, A good universal weight for multiple recurrence averages with commuting transformations in norm, Ergodic Theory Dynam. Systems, 37 (2017), 1009–1025, available at https://arXiv.org/abs/1506.06730. doi: 10.1017/etds.2015.76. Google Scholar

[6]

I. Assani and K. Presser, A survey of the return times theorem, in Ergodic Theory and Dynamical Systems, De Gruyter Proc. Math., De Gruyter, Berlin, 2014, 19–58. Google Scholar

[7]

J. P. Bell, M. Coons and K. G. Hare, The minimal growth of a k-regular sequence, Bull. Aust. Math. Soc., 90 (2014), 195–203. doi: 10.1017/S0004972714000197. Google Scholar

[8]

J. P. Bell, M. Coons and K. G. Hare, Growth degree classification for finitely generated semigroups of integer matrices, Semigroup Forum, 92 (2016), 23–44. doi: 10.1007/s00233-015-9725-1. Google Scholar

[9]

A. Bellow and V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, Trans. Amer. Math. Soc., 288 (1985), 307–345, URL http://dx.doi.org/10.2307/2000442. doi: 10.1090/S0002-9947-1985-0773063-8. Google Scholar

[10]

D. Berend, M. Lin, J. Rosenblatt and A. Tempelman, Modulated and subsequential ergodic theorems in Hilbert and Banach spaces, Ergodic Theory Dynam. Systems, 22 (2002), 1653–1665. doi: 10.1017/S0143385702000846. Google Scholar

[11]

J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Publ. Math., Inst. Hautes Études Sci., 69 (1985), 5–45, URL http://www.numdam.org/item?id=PMIHES_1989__69__5_0, With an appendix by the author, H. Furstenberg, Y. Katznelson and D. S. Ornstein. Google Scholar

[12]

J. Bourgain, H. Furstenberg, Y. Katznelson and D. S. Ornstein, Appendix on return-time sequences, Publ. Math., Inst. Hautes Études Sci., 69 (1985), 42–45, URL http://www.numdam.org/item?id=PMIHES_1989__69__42_0. Google Scholar

[13]

Z. Buczolich and R. D. Mauldin, Divergent square averages, Ann. of Math. (2), 171 (2010), 1479–1530. doi: 10.4007/annals.2010.171.1479. Google Scholar

[14]

Q. Chu, Convergence of weighted polynomial multiple ergodic averages, Proc. Amer. Math. Soc., 137 (2009), 1363–1369. doi: 10.1090/S0002-9939-08-09614-7. Google Scholar

[15]

D. Cömez, M. Lin and J. Olsen, Weighted ergodic theorems for mean ergodic L1- contractions, Trans. Amer. Math. Soc., 350 (1998), 101–117. doi: 10.1090/S0002-9947-98-01986-2. Google Scholar

[16]

C. Cuny and M. Weber, Ergodic theorems with arithmetical weights, Israel J. Math., 217 (2017), 139-180. doi: 10.1007/s11856-017-1441-y. Google Scholar

[17]

S. Drappeau and C. Müllner, Exponential sums with automatic sequences, 2017, Preprint, available at https://arXiv.org/abs/1710.01091.Google Scholar

[18]

M. Drmota and J. F. Morgenbesser, Generalized Thue-Morse sequences of squares, Israel J. Math., 190 (2012), 157–193. doi: 10.1007/s11856-011-0186-2. Google Scholar

[19]

P. Dumas, Joint spectral radius, dilation equations, and asymptotic behavior of radix-rational sequences, Linear Algebra Appl., 438 (2013), 2107–2126. doi: 10.1016/j.laa.2012.10.013. Google Scholar

[20]

P. Dumas, Asymptotic expansions for linear homogeneous divide-and-conquer recurrences: algebraic and analytic approaches collated, Theoret. Comput. Sci., 548 (2014), 25–53. doi: 10.1016/j.tcs.2014.06.036. Google Scholar

[21]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, vol. 259 of Graduate Texts in Mathematics, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2. Google Scholar

[22]

T. Eisner, A polynomial version of Sarnak's conjecture, C. R. Math. Acad. Sci. Paris, 353 (2015), 569–572. doi: 10.1016/j.crma.2015.04.009. Google Scholar

[23]

T. Eisner, Linear sequences and weighted ergodic theorems, Abstr. Appl. Anal., (2013), Art. ID 815726, 5 pp. Google Scholar

[24]

T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, vol. 272 of Graduate Texts in Mathematics, Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2. Google Scholar

[25]

T. Eisner and B. Krause, (Uniform) convergence of twisted ergodic averages, Ergodic Theory Dynam. Systems, 36 (2016), 2172–2202. doi: 10.1017/etds.2015.6. Google Scholar

[26]

T. Eisner and P. Zorin-Kranich, Uniformity in the Wiener-Wintner theorem for nilsequences, Discrete Contin. Dyn. Syst., 33 (2013), 3497–3516. doi: 10.3934/dcds.2013.33.3497. Google Scholar

[27]

E. H. El Abdalaoui, J. Ku laga-Przymus, M. Lemańczyk and T. de la Rue, The Chowla and the Sarnak conjectures from ergodic theory point of view, Discrete Contin. Dyn. Syst., 37 (2017), 2899–2944. doi: 10.3934/dcds.2017125. Google Scholar

[28]

A.-H. Fan, Weighted Birkhoff ergodic theorem with oscillating weights, Ergodic Theory and Dynamical Systems, (2017), 1-15. doi: 10.1017/etds.2017.81. Google Scholar

[29]

N. Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061–1071. doi: 10.1017/S0143385706000204. Google Scholar

[30]

A. O. Gel'fond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., 13 (1967/1968), 259-265. doi: 10.4064/aa-13-3-259-265. Google Scholar

[31]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2), 175 (2012), 465–540. doi: 10.4007/annals.2012.175.2.2. Google Scholar

[32]

B. Host and B. Kra, Uniformity seminorms on $ \ell^∞$ and applications, J. Anal. Math., 108 (2009), 219–276. doi: 10.1007/s11854-009-0024-1. Google Scholar

[33]

J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, 2017, Preprint, available at https://arXiv.org/abs/1611.09985.Google Scholar

[34]

B. Krause and P. Zorin-Kranich, A random pointwise ergodic theorem with Hardy field weights, Illinois J. Math., 59 (2015), 663–674, URL http://projecteuclid.org/euclid.ijm/1475266402. Google Scholar

[35]

P. LaVictoire, Universally L1-bad arithmetic sequences, J. Anal. Math., 113 (2011), 241–263. doi: 10.1007/s11854-011-0006-y. Google Scholar

[36]

E. Lesigne, Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener-Wintner, Ergodic Theory Dynam. Systems, 10 (1990), 513– 521. doi: 10.1017/S014338570000571X. Google Scholar

[37]

E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784. Google Scholar

[38]

E. Lesigne and C. Mauduit, Propriétés ergodiques des suites q-multiplicatives, Compositio Math., 100 (1996), 131–169, URL http://www.numdam.org/item?id=CM_1996__100_2_131_0. Google Scholar

[39]

E. Lesigne, C. Mauduit and B. Mossé, Le théorème ergodique le long d'une suite q-multiplicative, Compositio Math., 93 (1994), 49–79, URL http://www.numdam.org/item?id=CM_1994__93_1_49_0. Google Scholar

[40]

M. Lin, J. Olsen and A. Tempelman, On modulated ergodic theorems for Dunford-Schwartz operators, in Proceedings of the Conference on Probability, Ergodic Theory, and Analysis (Evanston, IL, 1997), 43 (1999), 542–567, URL http://projecteuclid.org/euclid.ijm/1255985110. Google Scholar

[41]

B. Martin, C. Mauduit and J. Rivat, Théorème des nombres premiers pour les fonctions digitales, Acta Arith., 165 (2014), 11–45. doi: 10.4064/aa165-1-2. Google Scholar

[42]

C. Mauduit, Automates finis et ensembles normaux, Ann. Inst. Fourier (Grenoble), 36 (1986), 1–25, URL http://www.numdam.org/item?id=AIF_1986__36_2_1_0. doi: 10.5802/aif.1044. Google Scholar

[43]

C. Mauduit, Propriétés arithmétiques des substitutions et automates infinis, Ann. Inst. Fourier (Grenoble), 56 (2006), 2525–2549, URL http://aif.cedram.org/item?id=AIF_200__56_7_2525_0, Numération, pavages, substitutions. doi: 10.5802/aif.2248. Google Scholar

[44]

C. Mauduit and A. Sárközy, On finite pseudorandom binary sequences. Ⅱ. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences, a further construction, J. Number Theory, 73 (1998), 256–276. doi: 10.1006/jnth.1998.2286. Google Scholar

[45]

C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219–3290. doi: 10.1215/00127094-2017-0024. Google Scholar

[46]

C. Müllner, Exponential Sum Estimates and Fourier Analytic Methods for Digitally Based Dynamical Systems, PhD thesis, Technische Universit at Wien, 2017.Google Scholar

[47]

K. Petersen, Ergodic Theory, vol. 2 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1989, Corrected reprint of the 1983 original. Google Scholar

[48]

M. Queffélec, Substitution Dynamical Systems—Spectral Analysis, vol. 1294 of Lecture Notes in Mathematics, 2nd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6. Google Scholar

[49]

T. Tao, Poincaré's Legacies, Pages from Year two of a Mathematical blog. Part I, American Mathematical Society, Providence, RI, 2009. Google Scholar

[50]

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

[51]

N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415–426. doi: 10.2307/2371534. Google Scholar

[52]

M. Wierdl, Pointwise ergodic theorems along the prime numbers, Israel J. Math., 64 (1988), 315-336. doi: 10.1007/BF02882425. Google Scholar

[53]

P. Zorin-Kranich, A double return times theorem, Israel J. Math., 204 (2014), 85–96, available at https://arXiv.org/abs/1506.05748. doi: 10.1007/s11856-014-1112-1. Google Scholar

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