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Pointwise convergence of triple ergodic averages for mixing transformations
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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.
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Possibly relevant literature (found by the agent, not checked)
- N. Frantzikinakis, Some open problems on multiple ergodic averages (arXiv:1103.3808, v1 March 2011, v3 October 2016). Problem 11 (Section 5.1.3) asks about pointwise convergence of the averages with iterates $T^n, T^{2n}, T^{3n}$, noting it is "open even for weak mixing systems"; the main theorem addresses exactly this configuration under the mixing hypothesis.
- Mariusz Mirek, Tomasz Z. Szarek and James Wright, Oscillation inequalities in ergodic theory and analysis: one-parameter and multi-parameter perspectives (arXiv:2209.01309, 3 September 2022). Surveys quantitative oscillation methods for pointwise convergence, context for the manuscript's integrated oscillation bound used to establish convergence of the three-function averages.
- Borys Kuca, Joint ergodicity - 40 years on (arXiv:2603.18974, 19 March 2026). A recent survey of norm and pointwise questions for multiple ergodic averages, including product-of-integrals limits, context for the manuscript's separate steps of proving pointwise convergence and identifying the limit; the paper discusses the unrestricted three-function problem via Kosz–Mirek–Peluse–Wan–Wright but does not cite this survey.
Digests and expositions
- Latent Seminar, Pointwise multiple ergodic averages under mixing (6 October 2026). Explains the four manuscripts' distinct scopes, oscillation estimates, selected-length constructions and companion dependencies, without claiming independent verification of the complete proofs. [family-level]
Elsewhere
- Emergent Mind: family 154
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026,
author = {{OpenAI}},
title = {{Pointwise convergence of triple ergodic averages for mixing transformations}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf}{OAI:Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026}},
year = {2026}
}Family 154: Pointwise multiple ergodic averages for mixing transformations
Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.
Other papers in this family:
- Pointwise Multiple Ergodic Averages for Mixing Transformations · October 4, 2026
- Pointwise convergence of fourfold ergodic averages for mixing transformations · October 4, 2026
- Triple ergodic averages with distinct integer slopes · October 4, 2026