Convergence of ergodic averages for many group rotations

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^αf(x). The f-rotation set is Gamma_f={α\in G: M_N^α 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} α)/k, k=1,... for a.e. x for many α, hence some of our theorems are stated by using instead of Gamma_f slightly larger sets, denoted by Gamma_{f,b}.