Quantitative Convergence for Sparse Ergodic Averages in $L^1$

Abstract

We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that \[ a_n \in \{ \lfloor n^c \rfloor, \min\{ k : \sum_{j \leq k} X_j = n\} \} \] where $X_j$ are Bernoulli random variables with expectations $\mathbb{E}{X}_j=n^{-\hat{I}\pm }$, and we restrict to $1<c<7/6,0<\hat{I}\pm <1/2$. Then (almost surely) for any measure-preserving system, $(X,\hat{I}¼,T)$, and any $f\in L^1(X)$, the ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{a_n} f \] converge $\hat{I}¼$-a.e. Moreover, our proof gives new quantitative estimates on the rate of convergence, using jump-counting/variation/oscillation technology, pioneered by Bourgain. This improves on previous work of Urban-Zienkiewicz, and Mirek, who established the above with $c=\frac{1001}{1000},\frac{30}{29}$, respectively, and LaVictoire, who established the random result, all in a non-quantitative setting.