Pointwise triple ergodic averages with distinct integer slopes under mixing
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Definitions and mathematical statement
Statement recorded in the supplied source.
For every invertible mixing probability-preserving transformation T, every three pairwise distinct nonzero integers a,b,c, and bounded measurable f,g,h, do $(1/N)sum_{n=1}^N$ $f(T^{an}x)g(T^{bn}x)h(T^{cn}x)$ converge almost everywhere to (integral f)(integral g)(integral h), including mixed-sign slopes?Short English statement
For every invertible mixing probability-preserving transformation T, every three pairwise distinct nonzero integers a,b,c, and bounded measurable f,g,h, do $(1/N)sum_{n=1}^N$ $f(T^{an}x)g(T^{bn}x)h(T^{cn}x)$ converge almost everywhere to (integral f)(integral g)(integral h), including mixed-sign slopes?Sources
- S1 Original question source or mathematical context. https://arxiv.org/abs/1103.3808.
- S2 Original question source or mathematical context. https://doi.org/10.4171/owr/2023/25.
- S3 Original question source or mathematical context. https://doi.org/10.4171/owr/2025/51.
- S4 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2685
Manuscript 1 (family 154, 2026-10-04): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.
Pointwise triple ergodic averages with distinct integer slopes under mixing
OpenAI's claim
Triple averages with any three distinct nonzero integer slopes, including negative ones, are claimed to converge almost everywhere to the product of integrals for mixing transformations. The cited manuscript argues for that result; its relation to this problem is specified below.Mathematical statement of the claimed result
Mathematical claim 1.
For an invertible bimeasurable probability-preserving mixing $T$, any pairwise distinct nonzero integers $\lambda_1,\lambda_2,\lambda_3$ and fixed bounded measurable complex functions $f_j$, OpenAI claims \[ \frac1N\sum_{k=1}^N\prod_{j=1}^3 f_j(T^{\lambda_j k}x) \longrightarrow\prod_{j=1}^3\int_X f_j\,d\mu \quad\mu\text{-almost everywhere}. \] Mixing means the two-set correlation condition as $|m|\to\infty$. Negative slopes are allowed, convergence is through all positive $N$, and the excluded null set is allowed to vary when the slopes or functions change.Outline of the argument
Manuscript 1 (family 154, 2026-10-04): The local progression estimates are extended using an integer left inverse of the complete primitive configuration. Vandermonde cancellation and a height base divisible by every slope difference preserve the Fourier and divisibility bounds through discretization, giving the slope-dependent oscillation estimate needed for convergence.Formal verification
Family 154 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.
Status caveat
OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.Sources
- OpenAI, Triple ergodic averages with distinct integer slopes, 2026-10-04; citation key
OAI:Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.