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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.","authors_text":"Idris Assani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-12-18T19:21:03Z","title":"Pointwise recurrence for commuting measure preserving transformations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5270","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0d24d8f1d59604511bce0e05c563540266b702adc28801025a7e031cf3b28496","target":"record","created_at":"2026-05-18T01:41:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9e3abe681033ca676a55f5d4f20fa4f1e0cf1d9720df0ea84f6ce3d1a157b443","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-12-18T19:21:03Z","title_canon_sha256":"dee735db9ce55ced211c7349a1d70e80933c4857cdb15e8028df2bcc2edd441a"},"schema_version":"1.0","source":{"id":"1312.5270","kind":"arxiv","version":2}},"canonical_sha256":"448750b3b851138f5eff4dc35694f10ad4018eed8c2b52220d35c1804c56ba72","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"448750b3b851138f5eff4dc35694f10ad4018eed8c2b52220d35c1804c56ba72","first_computed_at":"2026-05-18T01:41:00.199624Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:41:00.199624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/deY3j9FsPo5gNNNBQrc10zKZJxHvDy/DQGn14HQPiOrhgH8IZnpI/+V0p7HDZaq9NLIWN3w4Fz7vPVaA37aBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:41:00.200104Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.5270","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0d24d8f1d59604511bce0e05c563540266b702adc28801025a7e031cf3b28496","sha256:e9a6f4e0c30fd7ce82ff5f020cc1b4cb774a52aed8c232090c32d94ecc36888c"],"state_sha256":"fd112739574691abc0bcbd3e9fd10b706e14167621eaf82fa9bc528a6483a32a"}