{"paper":{"title":"A counterexample on multiple convergence without commutativity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Song Shao, Wen Huang, Xiangdong Ye","submitted_at":"2024-07-15T13:53:17Z","abstract_excerpt":"It is shown that there exist a probability space $(X,{\\mathcal X},\\mu)$, two ergodic measure preserving transformations $T,S$ acting on $(X,{\\mathcal X},\\mu)$ with $h_\\mu(X,T)=h_\\mu(X,S)=0$, and $f, g \\in L^\\infty(X,\\mu)$ such that the limit \\begin{equation*}\n  \\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{n=0}^{N-1} f(T^{n}x)g(S^{n}x) \\end{equation*} does not exist in $L^2(X,\\mu)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.10728","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.10728/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}