Pointwise Convergence for Subsequences of Weighted Averages

الأصل · EN

We prove that if μₙ are probability measures on Z such that μₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence {nₖ} such that the weighted ergodic averages corresponding to μₙₖ satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n²+ ρ(n) for a slowly growing function ρ. Under some monotonicity assumptions, the rate of growth of ρ'(x) determines the existence of a "good" subsequence of these averages.