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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 and Continuous Dynamical Systems, 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. [2] J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563. [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. [4] I. Assani, Wiener Wintner Ergodic Theorems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/4538. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] S. Drappeau and C. Müllner, Exponential sums with automatic sequences, 2017, Preprint, available at https://arXiv.org/abs/1710.01091. [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. [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. [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. [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. [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. [23] T. Eisner, Linear sequences and weighted ergodic theorems, Abstr. Appl. Anal., (2013), Art. ID 815726, 5 pp. [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. [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. [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. [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. [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. [29] N. Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061–1071. doi: 10.1017/S0143385706000204. [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. [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. [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. [33] J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, 2017, Preprint, available at https://arXiv.org/abs/1611.09985. [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. [35] P. LaVictoire, Universally L1-bad arithmetic sequences, J. Anal. Math., 113 (2011), 241–263. doi: 10.1007/s11854-011-0006-y. [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. [37] E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784. [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. [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. [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. [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. [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. [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. [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. [45] C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219–3290. doi: 10.1215/00127094-2017-0024. [46] C. Müllner, Exponential Sum Estimates and Fourier Analytic Methods for Digitally Based Dynamical Systems, PhD thesis, Technische Universit at Wien, 2017. [47] K. Petersen, Ergodic Theory, vol. 2 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1989, Corrected reprint of the 1983 original. [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. [49] T. Tao, Poincaré's Legacies, Pages from Year two of a Mathematical blog. Part I, American Mathematical Society, Providence, RI, 2009. [50] P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982. [51] N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415–426. doi: 10.2307/2371534. [52] M. Wierdl, Pointwise ergodic theorems along the prime numbers, Israel J. Math., 64 (1988), 315-336.  doi: 10.1007/BF02882425. [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.

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. [2] J.-P. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563. [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. [4] I. Assani, Wiener Wintner Ergodic Theorems, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/4538. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] S. Drappeau and C. Müllner, Exponential sums with automatic sequences, 2017, Preprint, available at https://arXiv.org/abs/1710.01091. [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. [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. [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. [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. [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. [23] T. Eisner, Linear sequences and weighted ergodic theorems, Abstr. Appl. Anal., (2013), Art. ID 815726, 5 pp. [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. [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. [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. [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. [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. [29] N. Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems, 26 (2006), 1061–1071. doi: 10.1017/S0143385706000204. [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. [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. [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. [33] J. Konieczny, Gowers norms for the Thue-Morse and Rudin-Shapiro sequences, 2017, Preprint, available at https://arXiv.org/abs/1611.09985. [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. [35] P. LaVictoire, Universally L1-bad arithmetic sequences, J. Anal. Math., 113 (2011), 241–263. doi: 10.1007/s11854-011-0006-y. [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. [37] E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems, 13 (1993), 767-784. [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. [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. [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. [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. [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. [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. [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. [45] C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Math. J., 166 (2017), 3219–3290. doi: 10.1215/00127094-2017-0024. [46] C. Müllner, Exponential Sum Estimates and Fourier Analytic Methods for Digitally Based Dynamical Systems, PhD thesis, Technische Universit at Wien, 2017. [47] K. Petersen, Ergodic Theory, vol. 2 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1989, Corrected reprint of the 1983 original. [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. [49] T. Tao, Poincaré's Legacies, Pages from Year two of a Mathematical blog. Part I, American Mathematical Society, Providence, RI, 2009. [50] P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982. [51] N. Wiener and A. Wintner, Harmonic analysis and ergodic theory, Amer. J. Math., 63 (1941), 415–426. doi: 10.2307/2371534. [52] M. Wierdl, Pointwise ergodic theorems along the prime numbers, Israel J. Math., 64 (1988), 315-336.  doi: 10.1007/BF02882425. [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.
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