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Multiple recurrence and convergence for hardy sequences of polynomial growth

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Abstract

We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for “typical” choices of Hardy field functions a(t) with polynomial growth, the averages

$${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[a(n)]}}x) \cdots {f_\ell }({T^{\ell [a(n)]}}x)} $$

converge in mean and we determine their limit. For example, this is the case if a(t) = t 3/2, t log t, or t 2 + (log t)2. Furthermore, if {a 1(t), …, a ℓ (t)} is a “typical” family of logarithmico-exponential functions of polynomial growth, then for every ergodic system, the averages

$${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[{a_1}(n)]}}x) \cdots {f_\ell }({T^{[{a_\ell }(n)]}}x)} $$

converge in mean to the product of the integrals of the corresponding functions. For example, this is the case if the functions a i (t) are given by different positive fractional powers of t. We deduce several results in combinatorics. We show that if a(t) is a non-polynomial Hardy field function with polynomial growth, then every set of integers with positive upper density contains arithmetic progressions of the form {m,m + [a(n)], …, m + ℓ[a(n)]}. Under suitable assumptions, we get a related result concerning patterns of the form {m,m + [a 1(n)], …,m + [a ℓ (n)]}.

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Authors and Affiliations

  1. Department of Mathematics, University of Crete, Knossos Avenue, Heraklion, 71409, Greece

    Nikos Frantzikinakis

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  1. Nikos Frantzikinakis
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Correspondence to Nikos Frantzikinakis.

Additional information

The author was partially supported by NSF grant DMS-0701027 and Marie Curie International Reintegration Grant 248008.

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Frantzikinakis, N. Multiple recurrence and convergence for hardy sequences of polynomial growth. JAMA 112, 79–135 (2010). https://doi.org/10.1007/s11854-010-0026-z

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  • DOI: https://doi.org/10.1007/s11854-010-0026-z

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