Convergence of ergodic averages for many group rotations

TL;DR

This paper investigates the convergence of nonconventional ergodic averages for group rotations, establishing conditions under which such convergence implies integrability of the function, with distinctions based on the structure of the dual group.

Contribution

It proves that for compact locally connected Abelian groups, positive measure of the rotation set implies the function is integrable, and explores differences in behavior for p-adic groups and groups with infinite torsion.

Findings

Convergence of averages implies f is in L^1 for certain groups.

Results differ when the dual group has infinite multiple torsion.

Boundedness of the tail of the sequence is crucial for convergence.

Abstract

Suppose that G is a compact Abelian topological group, m is the Haar measure on G and f is a measurable function. Given (n_k), a strictly monotone increasing sequence of integers we consider the nonconventional ergodic/Birkhoff averages M_N^{\alpha}f(x). The f-rotation set is Gamma_f={\alpha \in G: M_N^{\alpha} f(x) converges for m a.e. x as N\to \infty .} We prove that if G is a compact locally connected Abelian group and f: G -> R is a measurable function then from m(Gamma_f)>0 it follows that f \in L^1(G). A similar result is established for ordinary Birkhoff averages if G=Z_{p}, the group of p-adic integers. However, if the dual group, \hat{G} contains "infinitely many multiple torsion" then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, f(x+n_{k} {\alpha})/k, k=1,... for a.e. x for many \alpha, hence some of our theorems are stated by using instead of Gamma_f slightly larger sets, denoted by Gamma_{f,b}.