Polynomial ergodic averages for certain countable ring actions

TL;DR

This paper extends ergodic average results from integer actions to actions of locally compact abelian groups, with applications in combinatorics and dynamics, demonstrating convergence of polynomial ergodic averages in these broader settings.

Contribution

It generalizes Frantzikinakis's joint ergodicity conditions to second-countable locally compact abelian groups and applies this to polynomial averages in countable fields and rings.

Findings

Convergence of polynomial ergodic averages in countable field actions.

Extension of ergodic theorems to locally compact abelian group actions.

Applications to combinatorics and topological dynamics.

Abstract

A recent result of Frantzikinakis establishes sufficient conditions for joint ergodicity in the setting of $\mathbb{Z}$-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action $(T_n)_{n \in F}$ of a countable field $F$ with characteristic zero on a probability space $(X,\mathcal{B},\mu)$ and a family $\{p_1,\dots,p_k\}$ of independent polynomials, we have \[ \lim_{N \to \infty} \frac{1}{|\Phi_N|}\sum_{n \in \Phi_N} T_{p_1(n)}f_1\cdots T_{p_k(n)}f_k\ = \ \prod_{j=1}^k \int_X f_i \ d\mu,\] where $f_i \in L^{\infty}(\mu)$, $(\Phi_N)$ is a F{\o} lner sequence of $(F,+)$, and the convergence takes place in $L^2(\mu)$. This yields corollaries in combinatorics and topological dynamics. Second, we prove that a similar result holds for totally ergodic actions of suitable rings.