Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 154-04. Pointwise convergence of triple ergodic averages for mixing transformations
Manuscript map · lines 5355–5364
OpenAI@2026-10-07#rem:oai-p-154-04
Authors: OpenAI. Family 154.
Title: Pointwise convergence of triple ergodic averages for mixing transformations.
Abstract: We prove pointwise convergence of triple ergodic averages for every invertible mixing probability preserving transformation T of an arbitrary probability space $(X,{}mathcal F,{}mu)$, with measurable inverse. For every triple of bounded measurable functions $f_1,f_2,f_3:X{}to{}mathbb C$, ${}displaystyle {}frac1N{}sum_{n=1}{}N f_1(T{}nx)f_2(T{}{2n}x)f_3(T{}{3n}x) {}longrightarrow {}prod_{j=1}{}3{}int_X f_j{},d{}mu$ for μ-almost every x as $N{}to{}infty$ through all positive integers.
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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