{"paper":{"title":"Pointwise recurrence for commuting measure preserving transformations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Idris Assani","submitted_at":"2013-12-18T19:21:03Z","abstract_excerpt":"Let $(X,\\mathcal{A}, \\mu)$ be a probability measure space and let $T_i,$ $1\\leq i\\leq H,$ be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results;\n  The averages\n  $$\\frac{1}{N}\\sum_{n=1}^N f_1(T_1^nx)f_2(T_2^nx)\\cdots f_H(T_H^nx)$$ converge a.e. for every function $f_i \\in L^{\\infty}(\\mu)$ .\\\\ As a consequence if $T_i = T^i$ for $1\\leq i \\leq H$ where $T$ is an invertible measure preserving transformation on $(X, \\mathcal{A}, \\mu)$ then the averages\n  $$\\frac{1}{N}\\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)...f_H(T^{Hn}x)$$ converge a."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5270","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}