随机遍历平均在 \(L^1\) 端点处的逐点收敛

中文摘要 AI 辅助

在其极具影响力的论文《关于整数某些子集的最大遍历定理》中，Bourgain 引入了沿整数随机生成子集的遍历平均逐点收敛的研究，并为 \(p>1\) 建立了稳健的 \(L^p\) 理论。近期，LaVictoire 的工作在 \(L^1\) 端点处部分补充了这一理论，他处理了沿比平方数集合“稍密”的随机序列的平均，已知对这些序列普遍的 \(L^1\) 逐点收敛不成立。在本工作中，我们证明：几乎必然地，对每个保测系统和每个可积函数，沿具有二次增长的随机序列采样的遍历平均几乎处处收敛。我们的证明结合了 M. Christ 发展的调和分析方法与二次形式的“随机采样”，从而克服了主要的均匀性障碍。

英文摘要

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.

发表机构

• University of Rochester（罗切斯特大学）
• Princeton University（普林斯顿大学）
• University of Bristol（布里斯托大学）

机构由 AI 辅助整理，请以论文原文为准。