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The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4
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 |
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$.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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