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Pointwise Convergence of Random Ergodic Averages at the L1 Endpoint

  • Will Burstein
  • Lorenzo Catani
  • Ben Krause
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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 Lp-theory for p>1. More recently, this theory was partially complemented at the L1 endpoint by work of LaVictoire, who treated averages along random sequences that are ``slightly denser" than the set of squares, for which universal L1 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.

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Cite

BibTeXRIS
BibTeX
@misc{burstein2026pointwise,
  title = {{Pointwise Convergence of Random Ergodic Averages at the \(L^1\) Endpoint}},
  author = {Will Burstein and Lorenzo Catani and Ben Krause},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04106},
  eprint = {2610.04106},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04106},
}
RIS
TY  - UNPB
TI  - Pointwise Convergence of Random Ergodic Averages at the L^1 Endpoint
AU  - Will Burstein
AU  - Lorenzo Catani
AU  - Ben Krause
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.04106
UR  - https://arxiv.org/abs/2610.04106
AB  - 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.
ER  -