{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:RMBQ5ZWHOKEHIKY763CX2R7N7O","short_pith_number":"pith:RMBQ5ZWH","schema_version":"1.0","canonical_sha256":"8b030ee6c77288742b1ff6c57d47edfbb64d8c5c9a6cae4dbf0f2e755d3e09ac","source":{"kind":"arxiv","id":"1103.1819","version":4},"attestation_state":"computed","paper":{"title":"Lower Bounds on Ricci Curvature and Quantitative Behavior of Singular Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber, Jeff Cheeger","submitted_at":"2011-03-09T16:15:13Z","abstract_excerpt":"Let Y^n denote the Gromov-Hausdorff limit of a sequence M^n_i-> Y^n of v-noncollapsed riemannian manifolds with Ric_i\\geq-(n-1). The singular set S of Y has a stratification S^0\\subset S^1\\subset\\...\\subset S, where y\\in S^k if no tangent cone at y splits off a factor R^{k+1} isometrically. There is a known Hausdorff dimension bound dimS^k\\leq k. Here, we define for all \\eta>0, 0<r\\leq 1, the {\\it k-th effective singular stratum} S^k_{\\eta,r} such that \\bigcup_\\eta\\bigcap_r \\,\\cS^k_{\\eta,r}= \\cS^k. Sharpening the bound dim S^k\\leq k, we prove that the r-tubular neighborhood satisfies: Vol(T_r("},"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":"1103.1819","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-03-09T16:15:13Z","cross_cats_sorted":[],"title_canon_sha256":"63794de7102a42bd59cbae479cf17d26ce5ba96d3a6032f3b31595ff4a89b35e","abstract_canon_sha256":"f73564f4ec73b8a6e9122dbf9f9c77161c45753cf862d4932497f842735052bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:59.013170Z","signature_b64":"V5d53ivysBlZGt9F51F2dLkj/yD+twyX5gsnWCuR22YUYsIAo3iR4Q03zRu6XNjMAgQgkRcnDd+ZOs9szjQLAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b030ee6c77288742b1ff6c57d47edfbb64d8c5c9a6cae4dbf0f2e755d3e09ac","last_reissued_at":"2026-05-18T03:41:59.012555Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:59.012555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lower Bounds on Ricci Curvature and Quantitative Behavior of Singular Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber, Jeff Cheeger","submitted_at":"2011-03-09T16:15:13Z","abstract_excerpt":"Let Y^n denote the Gromov-Hausdorff limit of a sequence M^n_i-> Y^n of v-noncollapsed riemannian manifolds with Ric_i\\geq-(n-1). The singular set S of Y has a stratification S^0\\subset S^1\\subset\\...\\subset S, where y\\in S^k if no tangent cone at y splits off a factor R^{k+1} isometrically. There is a known Hausdorff dimension bound dimS^k\\leq k. Here, we define for all \\eta>0, 0<r\\leq 1, the {\\it k-th effective singular stratum} S^k_{\\eta,r} such that \\bigcup_\\eta\\bigcap_r \\,\\cS^k_{\\eta,r}= \\cS^k. Sharpening the bound dim S^k\\leq k, we prove that the r-tubular neighborhood satisfies: Vol(T_r("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1819","kind":"arxiv","version":4},"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":"1103.1819","created_at":"2026-05-18T03:41:59.012686+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.1819v4","created_at":"2026-05-18T03:41:59.012686+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1819","created_at":"2026-05-18T03:41:59.012686+00:00"},{"alias_kind":"pith_short_12","alias_value":"RMBQ5ZWHOKEH","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_16","alias_value":"RMBQ5ZWHOKEHIKY7","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_8","alias_value":"RMBQ5ZWH","created_at":"2026-05-18T12:26:41.206345+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/RMBQ5ZWHOKEHIKY763CX2R7N7O","json":"https://pith.science/pith/RMBQ5ZWHOKEHIKY763CX2R7N7O.json","graph_json":"https://pith.science/api/pith-number/RMBQ5ZWHOKEHIKY763CX2R7N7O/graph.json","events_json":"https://pith.science/api/pith-number/RMBQ5ZWHOKEHIKY763CX2R7N7O/events.json","paper":"https://pith.science/paper/RMBQ5ZWH"},"agent_actions":{"view_html":"https://pith.science/pith/RMBQ5ZWHOKEHIKY763CX2R7N7O","download_json":"https://pith.science/pith/RMBQ5ZWHOKEHIKY763CX2R7N7O.json","view_paper":"https://pith.science/paper/RMBQ5ZWH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.1819&json=true","fetch_graph":"https://pith.science/api/pith-number/RMBQ5ZWHOKEHIKY763CX2R7N7O/graph.json","fetch_events":"https://pith.science/api/pith-number/RMBQ5ZWHOKEHIKY763CX2R7N7O/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RMBQ5ZWHOKEHIKY763CX2R7N7O/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RMBQ5ZWHOKEHIKY763CX2R7N7O/action/storage_attestation","attest_author":"https://pith.science/pith/RMBQ5ZWHOKEHIKY763CX2R7N7O/action/author_attestation","sign_citation":"https://pith.science/pith/RMBQ5ZWHOKEHIKY763CX2R7N7O/action/citation_signature","submit_replication":"https://pith.science/pith/RMBQ5ZWHOKEHIKY763CX2R7N7O/action/replication_record"}},"created_at":"2026-05-18T03:41:59.012686+00:00","updated_at":"2026-05-18T03:41:59.012686+00:00"}