Powers of sequences and convergence of ergodic averages

A sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure-preserving system (X,Β,μ,T) and any bounded measurable function f, the averages (1/N) ΣNn=1f(Tsnx) converge in the L2(μ) norm. We construct a sequence (sn) which is good for the mean ergodic theorem but such that the sequence (s2n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (skn) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form (1/N) ΣNn=1f1(Tsnx)f2(T2snx) ⋯fℓ(Tℓsnx). We also prove a similar result for pointwise convergence of single ergodic averages. © 2009 Cambridge University Press