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If $X$ is distal, we prove that the average \\[\\frac{1}{N}\\sum_{i=0}^{N} f_1(T_1^nx)f_2(T_2^nx)\\cdots f_d(T_d^nx) \\] converges for $\\mu$-a.e. $x\\in X$ as $N\\to\\infty$. We also establish the pointwise convergence of averages along cubical configur"},"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":"1609.02529","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-09-08T18:49:04Z","cross_cats_sorted":[],"title_canon_sha256":"4b24ff358832aa44598864bc6c96c05d2d4f85035158ac89d61dabc604a9f507","abstract_canon_sha256":"f70566c8c82efc302502a43cdc1ac4d98cb2c38fb9d1f139d5b35206e14a2d1f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:54.417669Z","signature_b64":"Ingnpya0CCtrBYzkLX7CGvxHzmPUyv/VHhBdlQOdYur3dD5qiZTsOBuYL5sIqHeBGvHOay+H1RMEa45ObkOzAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d3746fe03e5f7b5f4cedb236c88492e2b8a0eeb5cc5577848ef6fd7eafd2bd7","last_reissued_at":"2026-05-18T01:04:54.417141Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:54.417141Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pointwise convergence of some multiple ergodic averages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Sebasti\\'an Donoso, Wenbo Sun","submitted_at":"2016-09-08T18:49:04Z","abstract_excerpt":"We show that for every ergodic system $(X,\\mu,T_1,\\ldots,T_d)$ with commuting transformations, the average \\[\\frac{1}{N^{d+1}} \\sum_{0\\leq n_1,\\ldots,n_d \\leq N-1} \\sum_{0\\leq n\\leq N-1} f_1(T_1^n \\prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n \\prod_{j=1}^d T_j^{n_j}x)\\cdots f_d(T_d^n \\prod_{j=1}^d T_j^{n_j}x). \\] converges for $\\mu$-a.e. $x\\in X$ as $N\\to\\infty$. If $X$ is distal, we prove that the average \\[\\frac{1}{N}\\sum_{i=0}^{N} f_1(T_1^nx)f_2(T_2^nx)\\cdots f_d(T_d^nx) \\] converges for $\\mu$-a.e. $x\\in X$ as $N\\to\\infty$. 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