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Pointwise Multiple Ergodic Averages for Mixing Transformations
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Abstract
Let T be an invertible mixing probability-preserving transformation. For every integer n ≥ 2 and every fixed tuple of bounded measurable functions, we prove that the consecutive multiple ergodic averages of length n converge almost everywhere to the product of the integrals, as the averaging length tends to infinity through all positive integers. The probability space need not be standard, and no rate of mixing is required.
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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"; Theorem 1.1 addresses this question for mixing $T$ and every length.
- V. V. Ryzhikov, Multiple mixing, 75 years of Rokhlin's problem (arXiv:2411.07234, November 2024). A survey of Rokhlin's question whether mixing implies $k$-fold mixing; the paper uses a positive answer to that question as an input, citing the companion manuscript on Rokhlin's multiple-mixing problem.
- Rene Rühr and Ronggang Shi, Quantitative Multiple pointwise convergence and effective multiple correlations (arXiv:1904.13055, 30 April 2019). Derives quantitative multiple pointwise ergodic theorems from effective higher-order correlations, a comparison point for this manuscript's convergence at every finite consecutive length without a mixing rate.
- 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 open problems, giving background on norm versus almost-everywhere convergence; the fourfold companion cites it, but this paper does not.
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
- VibeMathed: Problem record
- Emergent Mind: family 154
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026,
author = {{OpenAI}},
title = {{Pointwise Multiple Ergodic Averages for Mixing Transformations}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf}{OAI:Pointwise-Multiple-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 convergence of fourfold ergodic averages for mixing transformations · October 4, 2026
- Triple ergodic averages with distinct integer slopes · October 4, 2026
- Pointwise convergence of triple ergodic averages for mixing transformations · October 4, 2026