| Let (X, , P) be a non-atomic probability space and let T be an invertible measure-preserving transformation of ( X, , P). Fix a sequence (mk ) in and let f ∈ Lp( X), 1 ≤ p ≤ ∞. We know that, depending on what the powers are, the averages 1n k=1fTmk x may or may not converge a.e. x ∈ X, and they may or may not stay bounded a.e. We consider the properties of sequences (Ln) of real numbers and ( wn) of positive integers so that 1Ln k=1f Tmkx and 1Lnsup 1≤k≤n1k j=1 fTmjx converge a.e. x ∈ X for any sequence (mk) in . |
