Multiple recurrence and convergence for hardy sequences of polynomial growth
By Nikos Frantzikinakis. 2009
52 citations recorded by the source corpus.
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) = t3/2, t log t, or t2 + (log t)2. Furthermore, if {a1(t), …, aℓ(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 ai (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 + [a1(n)], …,m + [aℓ(n)]}.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/4a52e263209622a200d9cd245f5165a079c21eed
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In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
In Mathematical Physics
Citations in the corpus
- Szemeredi's theorem, frequent hypercyclicity and multiple recurrence (2010, background): “More general versions of this theorem can be found for example in [20], where the functions γ1, .”
- Some open problems on multiple ergodic averages (2011, background): “neralized polynomial sequences [44], and some random sequences of zero density [88]. The sequence ([nc]), where c ∈ R is positive, is known to be good for multiple recurrence of powers [90] (see also [79]), but if c ∈ R\Qis greater than 1, it is not known whether it is good for multiple recurrence o”