Pointwise convergence of multiple ergodic averages for mixing transformations
From OpenAI Aftermath
Pointwise convergence of multiple ergodic averages for mixing transformations concerns the almost-everywhere behaviour of the averages (1/N) Σk≤N f1(Tkx) ⋯ fn(Tnkx) along arithmetic progressions of times, for an invertible mixing measure-preserving transformation T and bounded functions fj. Four manuscripts dated 4 October 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim almost-everywhere convergence to the product of the integrals for every length n, with separate manuscripts for three and four terms and for three distinct integer slopes. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; this page does not certify them.
Background
- As recounted in the introductions of the manuscripts:
- Furstenberg's ergodic proof of Szemerédi's theorem made averages along arithmetic progressions of times central to ergodic Ramsey theory. Host and Kra proved convergence in L2 for every finite length on arbitrary probability-preserving systems, and Ziegler gave another proof. For weakly mixing transformations the norm limit is the product of the integrals by a classical van der Corput argument, and Bergelson's weakly mixing PET theorem includes distinct nonzero integer slopes of either sign.
- For pointwise convergence, Bourgain's double recurrence theorem treats two functions. Longer products were handled under additional structure: K-systems by Derrien and Lesigne, weakly mixing systems with singular spectrum on the Pinsker factor by Assani, weakly mixing systems whose pairwise independent self-joinings are independent by Gutman, Huang, Shao and Ye, and ergodic measure-distal systems, for every length, by Huang, Shao and Ye.
- The triple manuscript cites a September 2026 account by Kosz, Mirek, Peluse, Wan and Wright, whose theorem treats polynomials of distinct degrees, for a discussion of the unrestricted three-function problem at the times n, 2n, 3n.
The original results
- Claim (every length). Theorem thm:main of Pointwise Multiple Ergodic Averages for Mixing Transformations states that if T is an invertible mixing probability-preserving transformation, then for every n ≥ 2 and all bounded f1, …, fn the averages above converge almost everywhere to the product of the integrals, as N runs through all positive integers. The probability space need not be standard, and no rate of mixing is required.
- Claim (three and four terms). Pointwise convergence of triple ergodic averages for mixing transformations and Pointwise convergence of fourfold ergodic averages for mixing transformations state the same conclusion for the times n, 2n, 3n and n, 2n, 3n, 4n.
- Claim (distinct slopes). Triple ergodic averages with distinct integer slopes states the conclusion for the times λ1n, λ2n, λ3n, for pairwise distinct nonzero integers λj of either sign.
- Dependence on other manuscripts of the collection, as stated. The triple manuscript takes its analytic input from the collection's An L3 bound for the trilinear Hilbert transform (see The trilinear Hilbert transform), and the distinct-slopes manuscript extends the estimates of that manuscript and of the triple manuscript to general slopes. The fourfold and every-length manuscripts use the collection's affirmative answer to Rokhlin's problem (see Rokhlin's multiple-mixing problem). The every-length manuscript also uses an oscillation estimate for triple averages at distinct integer slopes that needs neither mixing nor ergodicity, and states that this estimate, with its profile approximation, yields pointwise convergence of triple averages at distinct positive integer slopes without a mixing assumption. The dependent manuscripts do not reproduce the companion proofs.
- Method, as described: the triple and distinct-slopes proofs establish an oscillation inequality for smooth dyadic averages in the style of Bourgain, from a local four-linear estimate whose inputs may vary with spatial depth. The fourfold and every-length proofs argue by contradiction: a failure of convergence gives selected-length laws on a suspension flow, a three-group channel structure is preserved through higher-order mixing and a tensor-packing estimate, and a saturated extension, related to Austin's sated extensions, produces a further output slot and contradicts maximality.
- Sources. At commit
1581d4c: Pointwise Multiple Ergodic Averages for Mixing Transformations, Pointwise convergence of triple ergodic averages for mixing transformations, Pointwise convergence of fourfold ergodic averages for mixing transformations and Triple ergodic averages with distinct integer slopes. - Formalization. The repository's Lean documentation has no scope note for this family at that commit.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 23 October 2026) |
|---|---|---|---|---|
| 2026-10-16 | J. Jardine, Lattice24, referee reviews rank_295 (every length), rank_443 (triple), rank_451 (fourfold) and rank_300 (distinct slopes) in JJardine919/openai-math-audit | Review (AI-assisted audit) | All four report no gap, with minor notes. Each states that its manuscript is conditional on companion manuscripts that it did not verify: the trilinear Hilbert transform estimate for the triple and distinct-slopes manuscripts, and the answer to Rokhlin's problem and the triple pointwise input on arbitrary systems for the fourfold and every-length manuscripts. The triple and fourfold reviews point out that the companion oscillation estimates do not use mixing, so that, if correct, they give pointwise convergence of the triple averages on every invertible system, a problem the triple review describes as open with reference to the literature the manuscript cites. The fourfold review states that its manuscript uses this consequence essentially and that the companion adaptations deserve the heaviest scrutiny in the family; it calls this a dependency risk, not a located gap. The every-length review also notes that a measurability step in its saturation lemma is only sketched. | Part of an audit of the collection's manuscripts that the audit describes as complete. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscripts.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
