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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1312.5270","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-12-18T19:21:03Z","cross_cats_sorted":[],"title_canon_sha256":"dee735db9ce55ced211c7349a1d70e80933c4857cdb15e8028df2bcc2edd441a","abstract_canon_sha256":"9e3abe681033ca676a55f5d4f20fa4f1e0cf1d9720df0ea84f6ce3d1a157b443"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:41:00.200104Z","signature_b64":"/deY3j9FsPo5gNNNBQrc10zKZJxHvDy/DQGn14HQPiOrhgH8IZnpI/+V0p7HDZaq9NLIWN3w4Fz7vPVaA37aBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"448750b3b851138f5eff4dc35694f10ad4018eed8c2b52220d35c1804c56ba72","last_reissued_at":"2026-05-18T01:41:00.199624Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:41:00.199624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1312.5270","created_at":"2026-05-18T01:41:00.199692+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.5270v2","created_at":"2026-05-18T01:41:00.199692+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.5270","created_at":"2026-05-18T01:41:00.199692+00:00"},{"alias_kind":"pith_short_12","alias_value":"ISDVBM5YKEJY","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_16","alias_value":"ISDVBM5YKEJY6XX7","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_8","alias_value":"ISDVBM5Y","created_at":"2026-05-18T12:27:49.015174+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ISDVBM5YKEJY6XX7JXBVNFHRBL","json":"https://pith.science/pith/ISDVBM5YKEJY6XX7JXBVNFHRBL.json","graph_json":"https://pith.science/api/pith-number/ISDVBM5YKEJY6XX7JXBVNFHRBL/graph.json","events_json":"https://pith.science/api/pith-number/ISDVBM5YKEJY6XX7JXBVNFHRBL/events.json","paper":"https://pith.science/paper/ISDVBM5Y"},"agent_actions":{"view_html":"https://pith.science/pith/ISDVBM5YKEJY6XX7JXBVNFHRBL","download_json":"https://pith.science/pith/ISDVBM5YKEJY6XX7JXBVNFHRBL.json","view_paper":"https://pith.science/paper/ISDVBM5Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.5270&json=true","fetch_graph":"https://pith.science/api/pith-number/ISDVBM5YKEJY6XX7JXBVNFHRBL/graph.json","fetch_events":"https://pith.science/api/pith-number/ISDVBM5YKEJY6XX7JXBVNFHRBL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ISDVBM5YKEJY6XX7JXBVNFHRBL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ISDVBM5YKEJY6XX7JXBVNFHRBL/action/storage_attestation","attest_author":"https://pith.science/pith/ISDVBM5YKEJY6XX7JXBVNFHRBL/action/author_attestation","sign_citation":"https://pith.science/pith/ISDVBM5YKEJY6XX7JXBVNFHRBL/action/citation_signature","submit_replication":"https://pith.science/pith/ISDVBM5YKEJY6XX7JXBVNFHRBL/action/replication_record"}},"created_at":"2026-05-18T01:41:00.199692+00:00","updated_at":"2026-05-18T01:41:00.199692+00:00"}