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Joint ergodicity of Hardy field sequences.(English)Zbl 1526.37009

The author studies the multiple ergodic averages \[\frac{1}{N}\sum_{n=1}^{N} f_1(T^{a_1(n)}x) \cdots f_k(T^{a_k(n)}x)\]for general sequences \(a_1(n),\dots,a_k(n)\) of integers, where \(T\) is an invertible measure-preserving map on a probability space \((X,\mathcal{X},\mu)\). Originated from the proof of Szemerédi’s theorem byH. Furstenberg [J. Anal. Math. 31, 204–256 (1977;Zbl 0347.28016)], which relies on ergodic theoretic tools, the mean convergence of the above averages has been established for a wide variety of sequences.
In this article, the main result is that if the sequences \(a_1, \dots,a_k\) arise from smooth functions of polynomial growth belonging to a Hardy field (see [G. H. Hardy, Proc. Lond. Math. Soc. (2) 10, 54–90 (1911;JFM 42.0437.02); Orders of infinity. The “Infinitärcalcül” of Paul du Bois-Reymond. 2nd edition. Cambridge: Cambridge University Press (1924;JFM 50.0153.04)]) and satisfy certain independence assumptions, then they are jointly ergodic. This means that the \(L^2\)-limit of the above averages exists and is equal to the product of the integrals of the functions \(f_1 , \dots, f_k\), whenever the underlying system \((X,\mathcal{X} ,\mu, T)\) is ergodic. Some typical examples of studied sequences are polynomial sequences with real coefficients, the sequences \([n^{3/2}]\), \([n\log n]\), \([n\log\log n + \exp (\sqrt{\log(n^2 + 1)})]\), \([n^{\sqrt{2}}/\log^2 n]\) and, in general, sequences arising from functions in the Hardy field \(\mathcal{LE}\) of logarithmic-exponential functions.
The main results of the paper also confirm a conjecture ofN. Frantzikinakis [Bull. Hell. Math. Soc. 60, 41–90 (2016;Zbl 1425.37004), Problem 23; J. Anal. Math. 112, 79–135 (2010;Zbl 1211.37008)], which is the content of Theorem 1.2. Furthermore, a partial answer to Problem 22 in [N. Frantzikinakis, loc. cit.], which asks for general convergence of averages of functions from a Hardy field, is given. In the case of weak mixing systems, one can relax the assumptions on the functions \(a_1 , \dots , a_k\) even further and establish a Furstenberg type weak mixing theorem, thus generalizing the results in [V. Bergelson, Ergodic Theory Dyn. Syst. 7, 337–349 (1987;Zbl 0645.28012)], and giving a positive answer to Problem 3 in [N. Frantzikinakis, loc. cit.].

MSC:

37A44 Relations between ergodic theory and number theory
37A30 Ergodic theorems, spectral theory, Markov operators
37A25 Ergodicity, mixing, rates of mixing
11B30 Arithmetic combinatorics; higher degree uniformity

Cite

References:

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[12]Frantzikinakis, Nikos, A multidimensional Szemer\'{e}di theorem for Hardy sequences of different growth, Trans. Amer. Math. Soc., 5653-5692 (2015) ·Zbl 1351.37038 ·doi:10.1090/S0002-9947-2014-06275-2
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[15]Frantzikinakis, Nikos, Joint ergodicity of fractional powers of primes, Forum Math. Sigma, Paper No. e30, 30 pp. (2022) ·Zbl 1529.37003 ·doi:10.1017/fms.2022.35
[16]Frantzikinakis, Nikos, A Hardy field extension of Szemer\'{e}di’s theorem, Adv. Math., 1-43 (2009) ·Zbl 1182.37007 ·doi:10.1016/j.aim.2009.03.017
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[25]Koutsogiannis, Andreas, Multiple ergodic averages for tempered functions, Discrete Contin. Dyn. Syst., 1177-1205 (2021) ·Zbl 1466.37004 ·doi:10.3934/dcds.2020314
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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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