Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 154-02. Pointwise convergence of fourfold ergodic averages for mixing transformations
Manuscript map · lines 5335–5344
OpenAI@2026-10-07#rem:oai-p-154-02
Authors: OpenAI. Family 154.
Title: Pointwise convergence of fourfold ergodic averages for mixing transformations.
Abstract: We prove that fourfold ergodic averages along the times $n,2n,3n,4n$ converge almost everywhere to the product of the integrals for every invertible, bimeasurable, mixing probability-preserving transformation and every fixed choice of bounded measurable inputs. Convergence holds along all positive integer averaging lengths. No quantitative mixing rate is assumed, and the probability space need not be standard.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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