{"id":"openai-math/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations","pdf":"https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf","url":"https://github.com/openai/math/tree/main/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026","tags":["ergodic-theory","dynamical-systems","measure-theory"],"score":0,"title":"Pointwise convergence of fourfold ergodic averages for mixing transformations","domain":"Dynamical systems and ergodic theory","family":"154","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DS","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-04","reactions":{},"formalized":false,"family_title":"Pointwise multiple ergodic averages for mixing transformations"}