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Pointwise convergence of fourfold ergodic averages for mixing transformations
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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.
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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 fourfold theorem concerns the mixing case of the analogous question with four iterates.
- 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 takes mixing of every finite order as an input from its 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 fourfold pointwise theorem under qualitative mixing alone.
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-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026,
author = {{OpenAI}},
title = {{Pointwise convergence of fourfold ergodic averages for mixing transformations}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf}{OAI:Pointwise-convergence-of-fourfold-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
- Triple ergodic averages with distinct integer slopes · October 4, 2026
- Pointwise convergence of triple ergodic averages for mixing transformations · October 4, 2026