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Triple ergodic averages with distinct integer slopes
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Abstract
For every invertible mixing probability-preserving transformation, triple ergodic averages of bounded measurable functions along any three pairwise distinct nonzero integer slopes converge almost everywhere to the product of their integrals. The slopes may be positive or negative.
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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 treats the mixing case for arbitrary distinct nonzero integer slopes.
- Rene Rühr and Ronggang Shi, Quantitative Multiple pointwise convergence and effective multiple correlations (arXiv:1904.13055, 30 April 2019). Relates effective higher-order correlations to quantitative multiple pointwise convergence, a comparison point for this manuscript's triple-slope convergence without quantitative mixing assumptions.
- 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 oscillation estimates as tools for pointwise convergence, methodological context for the manuscript's oscillation estimate and its transfer to orbit averages at distinct integer slopes.
- B. Kuca, Joint ergodicity - 40 years on (arXiv:2603.18974, March 2026). A recent survey of norm and pointwise questions for multiple ergodic averages, including averages along several integer slopes; this paper does not cite it.
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:Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026,
author = {{OpenAI}},
title = {{Triple ergodic averages with distinct integer slopes}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026/triple-ergodic-distinct-slopes.pdf}{OAI:Triple-ergodic-averages-with-distinct-integer-slopes-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
- Pointwise convergence of triple ergodic averages for mixing transformations · October 4, 2026