# Multiple recurrence and convergence for Hardy sequences of polynomial growth - Mathematics > Dynamical Systems

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Abstract: We study the limiting behavior of multiple ergodic averages involvingsequences of integers that satisfy some regularity conditions and havepolynomial growth. We show that for -typical- choices of Hardy field functions$at$ with polynomial growth, the averages $\frac{1}{N}\sum {n=1}^Nf 1T^{an}x\cdot

.\cdot f \ellT^{\ell an}x$ converge in the meanand we determine their limit. For example, this is the case if $at=t^{3-2},t\log{t},$ or $t^2+\log{t}^2$. Furthermore, if ${a 1t,

.,a \ellt}$ is a-typical- family of logarithmico-exponential functions of polynomial growth,then for every ergodic system, the averages $\frac{1}{N}\sum {n=1}^Nf 1T^{a 1n}x\cdot

.\cdot f \ellT^{a \elln}x$ converge in the meanto the product of the integrals of the corresponding functions. For example,this is the case if the functions $a it$ are given by different positivefractional powers of $t$. We deduce several results in combinatorics. We showthat if $at$ is a non-polynomial Hardy field function with polynomial growth,then every set of integers with positive upper density contains arithmeticprogressions of the form ${m,m+an,

.,m+\ellan}$. Under suitableassumptions we get a related result concerning patterns of the form ${m,m+a 1n,

., m+a \elln}.$

Author: ** Nikos Frantzikinakis**

Source: https://arxiv.org/