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We also show that if $(X,\\mathcal{X},\\mu,S,T)$ is a measurable distal system, the average \\[ \\frac{1}{N}\\sum_{i=0}^{N-1} f_1 (S^i x) f_2 (T^i x) \\] converges for $\\mu$-a.e. $x\\in X$ as $N\\to \\infty$ for $f_1,f_2\\in L^{\\infty}(\\mu)$."},"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":"1509.09310","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-09-30T19:42:52Z","cross_cats_sorted":[],"title_canon_sha256":"667e470a3060e7d74ddb168d5a6e3d58e21d1d4e2c66d9db73c86502ae3c2a1a","abstract_canon_sha256":"332640f0bb28ceb33b4eff96263b2cdb59ded57eec4f5cb3261037400b71ac36"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:07.514185Z","signature_b64":"GEeiHTBE7VqQiUJhPY0rHRZQhr5KuJRT+WdWgCXMmKacj6GiZJmfP6T55jhVBzKVIgMSgUdX722lMphJVTpCDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a98c27c070a59d496e2c9079fa2e8d9f4f6510cd41afe39d0f2fa2929ce3e26d","last_reissued_at":"2026-05-18T00:51:07.513593Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:07.513593Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pointwise multiple averages for systems with two commuting transformations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Sebastian Donoso, Wenbo Sun","submitted_at":"2015-09-30T19:42:52Z","abstract_excerpt":"We show that if $(X,\\mathcal{X},\\mu,S,T)$ is an ergodic measure preserving system with commuting transformations $S$ and $T$, then the average \\[\\frac{1}{N^3} \\sum_{i,j,k=0}^{N-1} f_0(S^j T^k x) f_1 (S^{i+j} T^k x) f_2 (S^j T^{i+k} x)\\] converges for $\\mu$-a.e. $x\\in X$ as $N\\to \\infty$ for $f_0,f_1, f_2\\in L^\\infty(\\mu)$. We also show that if $(X,\\mathcal{X},\\mu,S,T)$ is a measurable distal system, the average \\[ \\frac{1}{N}\\sum_{i=0}^{N-1} f_1 (S^i x) f_2 (T^i x) \\] converges for $\\mu$-a.e. $x\\in X$ as $N\\to \\infty$ for $f_1,f_2\\in L^{\\infty}(\\mu)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.09310","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":"1509.09310","created_at":"2026-05-18T00:51:07.513688+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.09310v2","created_at":"2026-05-18T00:51:07.513688+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.09310","created_at":"2026-05-18T00:51:07.513688+00:00"},{"alias_kind":"pith_short_12","alias_value":"VGGCPQDQUWOU","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"VGGCPQDQUWOUS3RM","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"VGGCPQDQ","created_at":"2026-05-18T12:29:44.643036+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/VGGCPQDQUWOUS3RMSB47ULUNT5","json":"https://pith.science/pith/VGGCPQDQUWOUS3RMSB47ULUNT5.json","graph_json":"https://pith.science/api/pith-number/VGGCPQDQUWOUS3RMSB47ULUNT5/graph.json","events_json":"https://pith.science/api/pith-number/VGGCPQDQUWOUS3RMSB47ULUNT5/events.json","paper":"https://pith.science/paper/VGGCPQDQ"},"agent_actions":{"view_html":"https://pith.science/pith/VGGCPQDQUWOUS3RMSB47ULUNT5","download_json":"https://pith.science/pith/VGGCPQDQUWOUS3RMSB47ULUNT5.json","view_paper":"https://pith.science/paper/VGGCPQDQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.09310&json=true","fetch_graph":"https://pith.science/api/pith-number/VGGCPQDQUWOUS3RMSB47ULUNT5/graph.json","fetch_events":"https://pith.science/api/pith-number/VGGCPQDQUWOUS3RMSB47ULUNT5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VGGCPQDQUWOUS3RMSB47ULUNT5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VGGCPQDQUWOUS3RMSB47ULUNT5/action/storage_attestation","attest_author":"https://pith.science/pith/VGGCPQDQUWOUS3RMSB47ULUNT5/action/author_attestation","sign_citation":"https://pith.science/pith/VGGCPQDQUWOUS3RMSB47ULUNT5/action/citation_signature","submit_replication":"https://pith.science/pith/VGGCPQDQUWOUS3RMSB47ULUNT5/action/replication_record"}},"created_at":"2026-05-18T00:51:07.513688+00:00","updated_at":"2026-05-18T00:51:07.513688+00:00"}