OpenAI · 154-04 · 2026-10-04 · Dynamical systems and ergodic theory

Pointwise convergence of triple ergodic averages for mixing transformations

preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026

Family 154. Pointwise multiple ergodic averages for mixing transformations.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

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.

We prove pointwise convergence of triple ergodic averages for every invertible mixing probability preserving transformation T of an arbitrary probability space $(X, F, μ)$ , with measurable inverse. For every triple of bounded measurable functions $f_{1}, f_{2}, f_{3} : X \to C$ , $\frac{1}{N} n = 1 \sum^{N} f_{1} (T^{n} x) f_{2} (T^{2 n} x) f_{3} (T^{3 n} x) ⟶ j = 1 \prod^{3} \int_{X} f_{j} d μ$ for μ-almost every x as $N \to \infty$ through all positive integers.

Preprint source and Lean scope

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