Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications.(English)Zbl 1543.37006
This paper is a further contribution to understanding how far (in terms of diversity of functions and in particular the possible growth rates of the functions along which the dynamics is sampled) multiple recurrence phenomena extend. A particular feature is that multiple recurrence and an \(L^2\) ergodic theorem are shown for functions combining polynomials and non-polynomial powers. In the more general setting of functions chosen from a Hardy field a very general multiple recurrence phenomenon is shown. These ergodic statements imply new combinatorial results generalizing known extensions of Szemerédi’s theorem to polynomial and (certain) non-polynomial settings which are drawn out. Both examples and counter-examples indicating similar settings in which multiple recurrence cannot hold are also described. The results resolve some of the conjectures ofN. Frantzikinakis [Bull. Hell. Math. Soc. 60, 41–90 (2016;Zbl 1425.37004)].
Reviewer: Thomas B. Ward (Durham)
MSC:
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 37A44 | Relations between ergodic theory and number theory |
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
Citations:
Zbl 1425.37004Cite
References:
| [1] | Auslander, L.; Green, L.; Hahn, F., Flows on Homogeneous Spaces, Annals of Mathematics Studies, vol. 53, 1963, Princeton University Press: Princeton University Press Princeton, N.J., vii+107 ·Zbl 0099.39103 |
| [2] | Berend, D.; Bergelson, V., Jointly ergodic measure-preserving transformations, Isr. J. Math., 49, 4, 307-314, 1984 ·Zbl 0571.28012 |
| [3] | Beiglböck, M.; Bergelson, V.; Hindman, N.; Strauss, D., Some new results in multiplicative and additive Ramsey theory, Trans. Am. Math. Soc., 360, 2, 819-847, 2008 ·Zbl 1136.05074 |
| [4] | Bergelson, V., Ergodic Ramsey theory, (Logic and Combinatorics. Logic and Combinatorics, Arcata, Calif., 1985. Logic and Combinatorics. Logic and Combinatorics, Arcata, Calif., 1985, Contemp. Math., vol. 65, 1987, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 63-87 ·Zbl 0642.10052 |
| [5] | Bergelson, V., Ergodic Ramsey theory – an update, (Ergodic Theory of \(\mathbb{Z}^d\) Actions, Warwick, 1993-1994. Warwick, 1993-1994, London Math. Soc. Lecture Note Ser., vol. 228, 1996, Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 1-61 ·Zbl 0846.05095 |
| [6] | Bergelson, V.; Håland Knutson, I. J., Weak mixing implies weak mixing of higher orders along tempered functions, Ergod. Theory Dyn. Syst., 29, 5, 1375-1416, 2009 ·Zbl 1184.37009 |
| [7] | Bergelson, V.; Leibman, A., Polynomial extensions of van der Waerden’s and Szemerédi’s theorems, J. Am. Math. Soc., 9, 3, 725-753, 1996 ·Zbl 0870.11015 |
| [8] | Bergelson, V.; Leibman, A.; Lesigne, E., Intersective polynomials and the polynomial Szemerédi theorem, Adv. Math., 219, 1, 369-388, 2008 ·Zbl 1156.11007 |
| [9] | Bergelson, V.; Leibman, A.; Son, Y., Joint ergodicity along generalized linear functions, Ergod. Theory Dyn. Syst., 36, 7, 2044-2075, 2006 ·Zbl 1370.37006 |
| [10] | Bergelson, V.; Moreira, J.; Richter, F. K., Single and multiple recurrence along non-polynomial sequences, Adv. Math., 368, Article 107146 pp., 2020 ·Zbl 1441.37004 |
| [11] | Boshernitzan, M., An extension of Hardy’s class L of “orders of infinity”, J. Anal. Math., 39, 235-255, 1981 ·Zbl 0539.26002 |
| [12] | Boshernitzan, M., “Orders of infinity” generated by difference equations, Am. J. Math., 106, 5, 1067-1089, 1984 ·Zbl 0602.26002 |
| [13] | Boshernitzan, M., Uniform distribution and Hardy fields, J. Anal. Math., 62, 225-240, 1994 ·Zbl 0804.11046 |
| [14] | Chu, Q.; Frantzikinakis, N.; Host, B., Ergodic averages of commuting transformations with distinct degree polynomial iterates, Proc. Lond. Math. Soc. (3), 102, 5, 801-842, 2011 ·Zbl 1218.37009 |
| [15] | Corwin, L. J.; Greenleaf, F. P., Representations of Nilpotent Lie Groups and Their Applications. Part I: Basic Theory and Examples, Cambridge Studies in Advanced Mathematics, vol. 18, 1990, Cambridge University Press: Cambridge University Press Cambridge, viii+269 ·Zbl 0704.22007 |
| [16] | Donoso, S.; Koutsogiannis, A.; Sun, W., Seminorms for multiple averages along polynomials and applications to joint ergodicity, J. Anal. Math., 2021 |
| [17] | Frantzikinakis, N.; Kra, B., Ergodic averages for independent polynomials and applications, J. Lond. Math. Soc. (2), 74, 1, 131-142, 2006 ·Zbl 1099.37003 |
| [18] | Frantzikinakis, N., Equidistribution of sparse sequences on nilmanifolds, J. Anal. Math., 109, 353-395, 2009 ·Zbl 1186.37010 |
| [19] | Frantzikinakis, N., Multiple recurrence and convergence for Hardy sequences of polynomial growth, J. Anal. Math., 112, 79-135, 2010 ·Zbl 1211.37008 |
| [20] | Frantzikinakis, N., A multidimensional Szemerédi theorem for Hardy sequences of different growth, Trans. Am. Math. Soc., 367, 8, 5653-5692, 2015 ·Zbl 1351.37038 |
| [21] | Frantzikinakis, N., Some open problems on multiple ergodic averages, Bull. Hellenic Math. Soc., 60, 41-90, 2016 ·Zbl 1425.37004 |
| [22] | Frantzikinakis, N., Joint ergodicity of sequences, Adv. Math., 417, Article 108918 pp., 2023 ·Zbl 1514.37019 |
| [23] | Furstenberg, H., Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math., 31, 204-256, 1977 ·Zbl 0347.28016 |
| [24] | Frantzikinakis, N.; Wierdl, M., A Hardy field extension of Szemerédi’s theorem, Adv. Math., 222, 1, 1-43, 2009 ·Zbl 1182.37007 |
| [25] | Grace, J. H., Note on a Diophantine approximation, Proc. Lond. Math. Soc. (2), 17, 316-319, 1918 ·JFM 47.0166.02 |
| [26] | Hardy, G. H., Properties of logarithmico-exponential functions, Proc. Lond. Math. Soc. (2), 10, 54-90, 1912 ·JFM 42.0437.02 |
| [27] | Hardy, G. H., Orders of Infinity. The Infinitärcalcül of Paul du Bois-Reymond, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 12, 1971, Hafner Publishing Co.: Hafner Publishing Co. New York, v+62 ·JFM 50.0153.04 |
| [28] | Host, B.; Kra, B., Convergence of polynomial ergodic averages, Isr. J. Math., 149, 1-19, 2005, Probability in mathematics ·Zbl 1085.28009 |
| [29] | Host, B.; Kra, B., Nonconventional ergodic averages and nilmanifolds, Ann. Math. (2), 161, 1, 397-488, 2005 ·Zbl 1077.37002 |
| [30] | Koutsogiannis, A., Integer part polynomial correlation sequences, Ergod. Theory Dyn. Syst., 38, 4, 1525-1542, 2018 ·Zbl 1393.37007 |
| [31] | Leibman, A., Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergod. Theory Dyn. Syst., 25, 1, 201-213, 2005 ·Zbl 1080.37003 |
| [32] | Leibman, A., Rational sub-nilmanifolds of a compact nilmanifold, Ergod. Theory Dyn. Syst., 26, 3, 787-798, 2006 ·Zbl 1095.37002 |
| [33] | Raghunathan, M. S., Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68, 1972, Springer-Verlag: Springer-Verlag New York-Heidelberg, ix+227 ·Zbl 0254.22005 |
| [34] | Richter, F. K., Uniform distribution in nilmanifolds along functions from a Hardy field, J. Anal. Math., 149, 421-483, 2023 ·Zbl 1522.37003 |
| [35] | Szemerédi, E., On sets of integers containing k elements in arithmetic progression, Acta Arith., 27, 1, 199-245, 1975, Eng. ·Zbl 0303.10056 |
| [36] | Tsinas, K., Joint ergodicity of Hardy field sequences, Trans. Am. Math. Soc., 376, 3191-3263, 2023 ·Zbl 1526.37009 |
| [37] | Weyl, H., Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann., 77, 3, 313-352, 1916 ·JFM 46.0278.06 |
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