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Pointwise convergence of multiple ergodic averages for mixing transformations

Let TT be an invertible measure-preserving transformation of a probability space and f1,…,fnf_1,\dots,f_n bounded measurable functions. Since Furstenberg's ergodic proof of Szemeredi's theorem, the multiple ergodic averages 1N∑k=1N∏j=1nfj(Tjkx)\frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x) have been central. Host-Kra and Ziegler proved convergence in L2L^2 for every nn, but almost-everywhere convergence was known only for n=2n=2 (Bourgain) and under extra structure for larger nn: K-systems, weakly mixing systems with spectral or joining restrictions, and distal systems (Huang-Shao-Ye). The general pointwise question asks whether these averages converge almost everywhere for every nn and every system; it was open even for mixing TT. Do the multiple ergodic averages converge almost everywhere for every finite length?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Ergodic theory; multiple ergodic averages
Posed by
The pointwise convergence question for multiple ergodic averages, arising from Furstenberg's ergodic Szemeredi theorem; surveyed by Kuca (2026)
Year posed
—
Years open
—
Solved
2026-10-04
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
40 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims Theorem 1.1: for every invertible mixing probability-preserving TT (no standardness, no mixing rate), every n≥2n\ge2 and bounded f1,…,fnf_1,\dots,f_n, 1N∑k≤N∏jfj(Tjkx)→∏j∫fj\frac1N\sum_{k\le N}\prod_j f_j(T^{jk}x)\to\prod_j\int f_j almost everywhere along all NN. Companions treat n=3,4n=3,4 and three distinct nonzero slopes of either sign for mixing TT. Lemma 2.5 of the principal manuscript also states almost-everywhere convergence of averages of at most three functions at distinct positive slopes on every standard system, derived from the companion's oscillation estimate. It does NOT settle the general question for n≥4n\ge4 on non-mixing systems, nor polynomial iterates.

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. All four manuscripts of the family (dated 4 October 2026) are credited to OpenAI with no human author named. The principal manuscript uses as inputs two other release results: Rokhlin's multiple-mixing theorem (a separate catalog entry) and an oscillation estimate from the distinct-slopes triple-average companion, which in turn rests on a release paper on the trilinear Hilbert transform.

Verification

No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript was read against the posed question: it is the mixing case only, for consecutive progressions k,2k,…,nkk,2k,\dots,nk, with the null set depending on the tuple. No Lean formalisation exists for this family. The proof depends on unformalised release results (Rokhlin's multiple-mixing theorem, catalog entry rokhlin-multiple-mixing-problem, and the triple-average oscillation estimate), so an error there would propagate. The release README warns that some unformalised results could have issues.

Sources

Changelog1 change

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