On the pointwise ergodic theorem onL ^p for arithmetic sets

Abstract

The purpose of this note is to show how the results of [B] on the pointwise ergodic theorem forL ²-functions may be extended toL ^p for certainp<2. More precisely, we give a proof of the almost sure convergence of the means$\frac{1}{N}\sum\limits_{1\underline \le n\underline \le N} {T^{(n1)} }$ (t≧1) given a dynamical system (Ω,B, μ, T) andf of classL ^p(Ω,μ),p>(√5+1)/2.