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Math-1977 · mathematics · Dynamical systems

Multiple ergodic averages — Problem 19

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $(X,\mathcal B,\mu)$ be a probability space, and let $T,S:X\to X$ be invertible measure-preserving transformations which commute. Equalities between measurable transformations may be understood modulo null sets. For bounded measurable complex functions $f,g$, define \[ A_N(x)=\frac1N\sum_{n=1}^N f(T^n x)g(S^n x). \] Prove that $A_N(x)$ converges to a finite complex number for $\mu$-almost every $x$, for every such system and every $f,g\in L^\infty(\mu)$.

The exceptional set may depend on $T,S,f,g$; there is no requirement of one common exceptional set for all bounded measurable functions. No ergodicity or mixing assumption is made on the individual maps or on their joint action. In particular, $S$ is not required to be a power of $T$.

The target concerns almost-everywhere convergence along all averaging lengths $N$, without passage to a subsequence. It is stronger than norm convergence in $L^2(\mu)$ and does not follow from such norm convergence alone. No formula for the limit is prescribed, and the product of the separate ergodic means is not assumed to be that limit. The commutation hypothesis is a substantive part of the question: the average combines the two orbits through the same starting point and the same time index, rather than averaging independently over two times.

Short English statement

Prove almost-everywhere convergence of the double average along two arbitrary commuting measure-preserving transformations.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1977-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1977

Manuscript 1 (family 154, 2026-10-04): Setting unused functions to one gives double convergence when the two commuting maps are distinct powers of a common mixing transformation. Arbitrary commuting transformations without that hypothesis remain outside the theorem.

Manuscript 2 (family 154, 2026-10-04): Setting unused functions to one gives double convergence when the two commuting maps are distinct powers of a common mixing transformation. Arbitrary commuting transformations without that hypothesis remain outside the theorem.

Manuscript 3 (family 154, 2026-10-04): Setting unused functions to one gives double convergence when the two commuting maps are distinct powers of a common mixing transformation. Arbitrary commuting transformations without that hypothesis remain outside the theorem.

Manuscript 4 (family 154, 2026-10-04): Setting unused functions to one gives double convergence when the two commuting maps are distinct powers of a common mixing transformation. Arbitrary commuting transformations without that hypothesis remain outside the theorem.

Multiple ergodic averages — Problem 19

OpenAI's claim

Manuscript 1 (family 154, 2026-10-04): For every invertible mixing transformation, consecutive multiple ergodic averages of any fixed finite length are claimed to have an almost-everywhere limit equal to the product of the input integrals, without a mixing rate; Manuscript 2 (family 154, 2026-10-04): Fourfold averages at times $n,2n,3n,4n$ are claimed to converge pointwise to the product of integrals for every invertible mixing system and bounded inputs on arbitrary probability spaces. Manuscript 3 (family 154, 2026-10-04): For any invertible mixing system, the threefold averages at $n,2n,3n$ are claimed to converge almost everywhere along all positive averaging lengths to the product of integrals; Manuscript 4 (family 154, 2026-10-04): 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.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $(X,\mathcal F,\mu)$ be any probability space with an invertible bimeasurable measure-preserving transformation $T$. Assume mixing, meaning $\mu(A\cap T^{-m}B)\to\mu(A)\mu(B)$ as $|m|\to\infty$ for all measurable $A,B$. OpenAI claims, for every $n\ge2$ and every fixed bounded measurable $f_1,\ldots,f_n$, \[ \frac1N\sum_{k=1}^N\prod_{j=1}^n f_j(T^{jk}x)\longrightarrow\prod_{j=1}^n\int_X f_j\,d\mu \quad\text{for almost every }x. \] The limit uses all positive integers $N$, requires no mixing rate or standardness of the probability space, and its null exceptional set may depend on the functions.
Mathematical claim 2.
For any probability space and any invertible bimeasurable measure-preserving mixing $T$, OpenAI claims for every fixed $f_1,\ldots,f_4\in L^\infty(\mu)$ that \[ \frac1N\sum_{k=1}^N\prod_{j=1}^4 f_j(T^{jk}x)\longrightarrow\prod_{j=1}^4\int_X f_j\,d\mu \quad\mu\text{-almost everywhere}. \] Mixing means $\mu(A\cap T^{-m}B)\to\mu(A)\mu(B)$ for every measurable $A,B$ as $|m|\to\infty$; no quantitative rate or standard-space assumption is imposed, and convergence is along all positive $N$.
Mathematical claim 3.
Take an arbitrary probability space and an invertible probability-preserving mixing $T$ with measurable inverse. For each fixed triple of bounded measurable complex functions, OpenAI claims \[ \frac1N\sum_{k=1}^N f_1(T^kx)f_2(T^{2k}x)f_3(T^{3k}x) \longrightarrow\prod_{j=1}^3\int_X f_j\,d\mu \quad\mu\text{-almost everywhere}. \] Mixing is the two-set condition $\mu(A\cap T^{-m}B)\to\mu(A)\mu(B)$ as $|m|\to\infty$. All positive $N$ are used and a different triple may have a different excluded null set.
Mathematical claim 4.
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): A minimal nonzero centered correlation makes smaller selected spatial lists independent. Conditional-expectation energy saturation stabilizes extensions, and a square joining array is regrouped to create an additional output-only slot; maximality of that slot count gives a contradiction using only the existing at-most-three-slot pointwise theorem. Manuscript 2 (family 154, 2026-10-04): Three-slot oscillation estimates and the companion multiple-mixing theorem supply the analytic inputs. Selected joining laws are packed into iterated arrays, where orthogonality bounds the number of extra-coordinate slots; saturated conditional laws and reanchoring increase that number, contradicting its maximal choice.

Manuscript 3 (family 154, 2026-10-04): The companion trilinear-Hilbert local construction is extended to inputs that change with spatial depth. Its four-term quadratic-phase cancellation yields a block oscillation bound growing like the square root of the block count; failure to be Cauchy would force linear growth, while mixing identifies the pointwise limit. Manuscript 4 (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

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