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Register a new userSeminorms for multiple averages along polynomials and applications to joint ergodicity
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Exploiting the recent work of Tao and Ziegler on the concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study the criteria of joint ergodicity for sequences of the form $(T^{p_{1,j}(n)}_{1}cdotldotscdot T^{p_{d,j}(n)}_{d})_{ninmathbb{Z}},$ $1leq jleq k$, where $T_{1},dots,T_{d}$ are commuting measure preserving transformations on a probability measure space and $p_{i,j}$ are integer polynomials. To be more precise, we provide a sufficient condition for such sequences to be jointly ergodic. We also give a characterization for sequences of the form $(T^{p(n)}_{i})_{ninmathbb{Z}}, 1leq ileq d$ to be jointly ergodic, answering a question due to Bergelson.
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For any measure preserving system $(X,mathcal{B},mu,T_1,ldots,T_d),$ where we assume no commutativity on the transformations $T_i,$ $1leq ileq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order $0$ and of Fejer functions, i.e., tempered functions of order $0.$ We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are in general bad for convergence on arbitrary systems, but they are good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.
We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure preserving $mathbb{Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a null sequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on $mathbb{Z}^{d}$-systems.
We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions $t^{3/2}, tlog t$ and $e^{sqrt{log t}}$. We show that if all non-trivial linear combinations of the functions $a_1,...,a_k$ stay logarithmically away from rational polynomials, then the $L^2$-limit of the ergodic averages $frac{1}{N} sum_{n=1}^{N}f_1(T^{lfloor{a_1(n)}rfloor}x)cdots f_k(T^{lfloor{a_k(n)}rfloor}x)$ exists and is equal to the product of the integrals of the functions $f_1,...,f_k$ in ergodic systems, which establishes a conjecture of Frantzikinakis. Under some more general conditions on the functions $a_1,...,a_k$, we also find characteristic factors for convergence of the above averages and deduce a convergence result for weak-mixing systems.
We study sets of recurrence, in both measurable and topological settings, for actions of $(mathbb{N},times)$ and $(mathbb{Q}^{>0},times)$. In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form ${(an+b)^{ell}/(cn+d)^{ell}: n in mathbb{N}}$ are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.
Generalized polynomials are mappings obtained from the conventional polynomials by the use of operations of addition, multiplication and taking the integer part. Extending the classical theorem of H. Weyl on equidistribution of polynomials, we show that a generalized polynomial $q(n)$ has the property that the sequence $(q(n) lambda)_{n in mathbb{Z}}$ is well distributed $bmod , 1$ for all but countably many $lambda in mathbb{R}$ if and only if $limlimits_{substack{|n| rightarrow infty n otin J}} |q(n)| = infty$ for some (possibly empty) set $J$ having zero density in $mathbb{Z}$. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of I. Vinogradov and G. Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.
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