Pointwise multiple ergodic averages for mixing transformations

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.

Manuscripts (4)

Pointwise Multiple Ergodic Averages for Mixing Transformations

October 4, 2026 · source · cite as \cite{OAI:Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026}

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

October 4, 2026 · source · cite as \cite{OAI:Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026}

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

October 4, 2026 · source · cite as \cite{OAI:Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026}

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

October 4, 2026 · source · cite as \cite{OAI:Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026}

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.

Result 154 · Dynamical systems and ergodic theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.