Strongly ergodic sequences of integers and the individual ergodic theorem

Abstract:

Let $S = \{ {k_1},{k_2}, \ldots \}$ be an increasing sequence of positive integers. We call $S$ strongly ergodic if for every measure preserving transformation $T$ on a probability space $(\Omega ,\mathcal {F},P)$ and every $f \in {L_1}(\Omega )$ we have ${\lim _{n \to \infty }}(1/n)\sum \nolimits _{j = 1}^n {f({T^{kj}}\omega ) = Pf(\omega )}$ a.e. where $Pf$ is the appropriate limit guaranteed by the individual ergodic theorem. We give sufficient conditions for a sequence $S$ to be strongly ergodic and provide examples.