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

We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that $$a_n\in \{\lfloor n^c\rfloor ,\min \{k:\sum _{j\le k}{X}_j=n\}\}$$ where $X_j$ are Bernoulli random variables with expectations $\mathbb{E}{X}_j=n^{-\alpha }$, and we restrict to $1<c<7\mathrm{/}6,0<\alpha <1\mathrm{/}2$. Then (almost surely) for any measure-preserving system, $(X,\mu ,T)$, and any $f\in L^1(X)$, the ergodic averages $$\frac{1}{N}\sum _{n\le N}{T}^{a_n}f$$ converge $\mu$-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.