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That is, we show that if $(X,\\mathcal{X},\\mu, T)$ is an ergodic distal system, and $f_1, \\ldots, f_d \\in L^{\\infty}(\\mu)$, then multiple ergodic averages $"},"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":"1406.5930","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-06-23T14:44:55Z","cross_cats_sorted":[],"title_canon_sha256":"9361ef1f11a0ad118b4f05b810323a997335542a730fefb78d12bfb849f1651b","abstract_canon_sha256":"3f8474ad78f74c619c89d3884c0c197d2a3c2be0d272ddc5b6a123d2b92f5796"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:44.679874Z","signature_b64":"bqWp+XbxBQgsqAQGeA6N+A1mh/YMSM6Gzuxc01EWJf4e3A/r9qaZLUnI5Z71hDKudOHtsgFN+S5pPw1lHf3hAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd50a314149b1d23f7d0229927f84e08d1a11fe9b561c592ba910c1b521aab05","last_reissued_at":"2026-05-18T00:42:44.679175Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:44.679175Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pointwise convergence of multiple ergodic averages and strictly ergodic models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Song Shao, Wen Huang, Xiangdong Ye","submitted_at":"2014-06-23T14:44:55Z","abstract_excerpt":"By building some suitable strictly ergodic models, we prove that for an ergodic system $(X,\\mathcal{X},\\mu, T)$, $d\\in{\\mathbb N}$, $f_1, \\ldots, f_d \\in L^{\\infty}(\\mu)$, the averages $$\\frac{1}{N^2} \\sum_{(n,m)\\in [0,N-1]^2} f_1(T^nx)f_2(T^{n+m}x)\\ldots f_d(T^{n+(d-1)m}x) $$ converge $\\mu$ a.e.\n  Deriving some results from the construction, for distal systems we answer positively the question if the multiple ergodic averages converge a.e. 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