{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:4M3VFYECGOQLFF52W7GPQSVDTC","short_pith_number":"pith:4M3VFYEC","schema_version":"1.0","canonical_sha256":"e33752e08233a0b297bab7ccf84aa398bbd7ec9d5e4779f40d3cafc06a64605b","source":{"kind":"arxiv","id":"1103.2113","version":1},"attestation_state":"computed","paper":{"title":"A note on Borel--Cantelli lemmas for non-uniformly hyperbolic dynamical systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"M. Nicol, N. Haydn, S. Vaienti, T. Persson","submitted_at":"2011-03-10T19:55:03Z","abstract_excerpt":"Let $(B_{i})$ be a sequence of measurable sets in a probability space $(X,\\mathcal{B}, \\mu)$ such that $\\sum_{n=1}^{\\infty} \\mu (B_{i}) = \\infty$. The classical Borel-Cantelli lemma states that if the sets $B_{i}$ are independent, then $\\mu (\\{x \\in X : x \\in B_{i} \\text{infinitely often (i.o.)}) = 1$. Suppose $(T,X,\\mu)$ is a dynamical system and $(B_i)$ is a sequence of sets in $X$. We consider whether $T^i x\\in B_i$ for $\\mu$ a.e.\\ $x\\in X$ and if so, is there an asymptotic estimate on the rate of entry. If $T^i x\\in B_i$ infinitely often for $\\mu$ a.e.\\ $x$ we call the sequence $B_i$ a Bor"},"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":"1103.2113","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-03-10T19:55:03Z","cross_cats_sorted":[],"title_canon_sha256":"623627c7114e13d65d74193db3adb0f0488a75166308c9f6cb00e038bfcb8914","abstract_canon_sha256":"566e5a3c92a13e065b057f2778b977424f73dc57c97be2b30eccd9cb6b731b02"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:02.947028Z","signature_b64":"G8XcA3OKPVO4QozLw1L8b+V1oDr/cRTC/QelpOB+fNKmyfy7KM030KRR8ZP/FjR0ngHovdhSydODaW6k7L9WBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e33752e08233a0b297bab7ccf84aa398bbd7ec9d5e4779f40d3cafc06a64605b","last_reissued_at":"2026-05-18T04:27:02.946441Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:02.946441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on Borel--Cantelli lemmas for non-uniformly hyperbolic dynamical systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"M. Nicol, N. Haydn, S. Vaienti, T. Persson","submitted_at":"2011-03-10T19:55:03Z","abstract_excerpt":"Let $(B_{i})$ be a sequence of measurable sets in a probability space $(X,\\mathcal{B}, \\mu)$ such that $\\sum_{n=1}^{\\infty} \\mu (B_{i}) = \\infty$. The classical Borel-Cantelli lemma states that if the sets $B_{i}$ are independent, then $\\mu (\\{x \\in X : x \\in B_{i} \\text{infinitely often (i.o.)}) = 1$. Suppose $(T,X,\\mu)$ is a dynamical system and $(B_i)$ is a sequence of sets in $X$. We consider whether $T^i x\\in B_i$ for $\\mu$ a.e.\\ $x\\in X$ and if so, is there an asymptotic estimate on the rate of entry. If $T^i x\\in B_i$ infinitely often for $\\mu$ a.e.\\ $x$ we call the sequence $B_i$ a Bor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2113","kind":"arxiv","version":1},"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":"1103.2113","created_at":"2026-05-18T04:27:02.946522+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.2113v1","created_at":"2026-05-18T04:27:02.946522+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.2113","created_at":"2026-05-18T04:27:02.946522+00:00"},{"alias_kind":"pith_short_12","alias_value":"4M3VFYECGOQL","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"4M3VFYECGOQLFF52","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"4M3VFYEC","created_at":"2026-05-18T12:26:20.644004+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/4M3VFYECGOQLFF52W7GPQSVDTC","json":"https://pith.science/pith/4M3VFYECGOQLFF52W7GPQSVDTC.json","graph_json":"https://pith.science/api/pith-number/4M3VFYECGOQLFF52W7GPQSVDTC/graph.json","events_json":"https://pith.science/api/pith-number/4M3VFYECGOQLFF52W7GPQSVDTC/events.json","paper":"https://pith.science/paper/4M3VFYEC"},"agent_actions":{"view_html":"https://pith.science/pith/4M3VFYECGOQLFF52W7GPQSVDTC","download_json":"https://pith.science/pith/4M3VFYECGOQLFF52W7GPQSVDTC.json","view_paper":"https://pith.science/paper/4M3VFYEC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.2113&json=true","fetch_graph":"https://pith.science/api/pith-number/4M3VFYECGOQLFF52W7GPQSVDTC/graph.json","fetch_events":"https://pith.science/api/pith-number/4M3VFYECGOQLFF52W7GPQSVDTC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4M3VFYECGOQLFF52W7GPQSVDTC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4M3VFYECGOQLFF52W7GPQSVDTC/action/storage_attestation","attest_author":"https://pith.science/pith/4M3VFYECGOQLFF52W7GPQSVDTC/action/author_attestation","sign_citation":"https://pith.science/pith/4M3VFYECGOQLFF52W7GPQSVDTC/action/citation_signature","submit_replication":"https://pith.science/pith/4M3VFYECGOQLFF52W7GPQSVDTC/action/replication_record"}},"created_at":"2026-05-18T04:27:02.946522+00:00","updated_at":"2026-05-18T04:27:02.946522+00:00"}