Pointwise multiple ergodic averages for mixing transformations

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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.

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The paper states

Theorem 1.1

In Pointwise convergence of triple ergodic averages for mixing transformations, 4 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Three other possible matches are listed below the card, among them Theorem 1.1 in Pointwise Multiple Ergodic Averages for Mixing Transformations and Theorem 1.1 in Pointwise convergence of fourfold ergodic averages for mixing transformations.

Let (X,\mathcal F,\mu) be an arbitrary probability space and let T:X\to X be an invertible mixing probability preserving transformation with measurable inverse. For every triple of bounded measurable functions f_1,f_2,f_3:X\to\C,

A_N(f_1,f_2,f_3)(x) \longrightarrow \prod_{j=1}^3\int_X f_j\,d\mu \quad\text{for $\mu$-almost every }x

as N\to\infty through all positive integers. The exceptional null set may depend on the triple of functions.

Three more possible matches

Theorem 1.1 in Pointwise Multiple Ergodic Averages for Mixing Transformations, 4 October 2026.
Theorem 1.1 in Pointwise convergence of fourfold ergodic averages for mixing transformations, 4 October 2026.
Theorem 1.1 in Triple ergodic averages with distinct integer slopes, 4 October 2026.

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Papers and explanations: Four current papers, each readable here as a page beside its PDF.

Papers

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Pointwise Multiple Ergodic Averages for Mixing Transformations

4 October 2026, current edition.

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Let T be an invertible mixing probability-preserving transformation. For every integer n ≥ 2 and every fixed tuple of bounded measurable functions, we prove that the consecutive multiple ergodic averages of length n converge almost everywhere to the product of the integrals, as the averaging length tends to infinity through all positive integers. The probability space need not be standard, and no rate of mixing is required.

Pointwise convergence of fourfold ergodic averages for mixing transformations

4 October 2026, current edition.

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We prove that fourfold ergodic averages along the times $n,2n,3n,4n$ converge almost everywhere to the product of the integrals for every invertible, bimeasurable, mixing probability-preserving transformation and every fixed choice of bounded measurable inputs. Convergence holds along all positive integer averaging lengths. No quantitative mixing rate is assumed, and the probability space need not be standard.

Triple ergodic averages with distinct integer slopes

4 October 2026, current edition.

Read it here PDF

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.

Pointwise convergence of triple ergodic averages for mixing transformations

4 October 2026, current edition.

Read it here PDF

We prove pointwise convergence of triple ergodic averages for every invertible mixing probability preserving transformation T of an arbitrary probability space $(X,\mathcal F,\mu)$, with measurable inverse. For every triple of bounded measurable functions $f_1,f_2,f_3:X\to\mathbb C$, $\displaystyle \frac1N\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)f_3(T^{3n}x) \longrightarrow \prod_{j=1}^3\int_X f_j\,d\mu$ for μ-almost every x as $N\to\infty$ through all positive integers.

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Papers elsewhere in the release that this family’s papers cite

Rokhlin’s multiple-mixing problem for one transformationCurrent edition; has an accompanying Lean statement Family Rokhlin’s multiple-mixing problemCited by Pointwise Multiple Ergodic Averages for Mixing Transformations and Pointwise convergence of fourfold ergodic averages for mixing transformations.
An L³ bound for the trilinear Hilbert transformCurrent edition; no accompanying Lean statement Family An L³ bound for the trilinear Hilbert transformCited by Pointwise convergence of fourfold ergodic averages for mixing transformations, Triple ergodic averages with distinct integer slopes and Pointwise convergence of triple ergodic averages for mixing transformations.

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