Pointwise multiple ergodic averages for mixing transformations

What OpenAI claims

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.

Papers (4)

1. Pointwise Multiple Ergodic Averages for Mixing Transformations

   4 October 2026 · Preprint, not peer reviewed · PDF

   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.

2. Pointwise convergence of fourfold ergodic averages for mixing transformations

   4 October 2026 · Preprint, not peer reviewed · PDF

   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.

3. Triple ergodic averages with distinct integer slopes

   4 October 2026 · Preprint, not peer reviewed · PDF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.

4. Pointwise convergence of triple ergodic averages for mixing transformations

   4 October 2026 · Preprint, not peer reviewed · PDF

   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 f₁,f₂,f₃:X → ℂ,

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.