Pointwise Convergence of Random Ergodic Averages at the $L^1$ Endpoint

Source abstract

In his highly influential paper, \emph{On the maximal ergodic theorem for certain subsets of the integers}, Bourgain introduced the study of pointwise convergence of ergodic averages along randomly generated subsets of the integers, developing a robust $L^p$-theory for $p>1$. More recently, this theory was partially complemented at the $L^1$ endpoint by work of LaVictoire, who treated averages along random sequences that are ``slightly denser" than the set of squares, for which universal $L^1$ pointwise convergence is known to fail. In this work, we prove that, almost surely, for every measure-preserving system and every integrable function, the ergodic averages formed by sampling along random sequences with quadratic growth converge almost everywhere. Our proof combines harmonic-analytic methods developed by M.~Christ with a ``random sampling" of quadratic forms, thereby overcoming the main uniformity obstruction.