{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:SLNOIM5WWZWA2YBSJZDZ4TDU4C","short_pith_number":"pith:SLNOIM5W","schema_version":"1.0","canonical_sha256":"92dae433b6b66c0d60324e479e4c74e090a9313856bdef91442c8d89b6f32818","source":{"kind":"arxiv","id":"1912.03615","version":1},"attestation_state":"computed","paper":{"title":"Quantitative Estimates on the Singular Sets of Alexandrov Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Aaron Naber, Nan Li","submitted_at":"2019-12-08T04:51:47Z","abstract_excerpt":"Let $X\\in\\text{Alex}\\,^n(-1)$ be an $n$-dimensional Alexandrov space with curvature $\\ge -1$. Let the $r$-scale $(k,\\epsilon)$-singular set $\\mathcal S^k_{\\epsilon,\\,r}(X)$ be the collection of $x\\in X$ so that $B_r(x)$ is not $\\epsilon r$-close to a ball in any splitting space $\\mathbb R^{k+1}\\times Z$. We show that there exists $C(n,\\epsilon)>0$ and $\\beta(n,\\epsilon)>0$, independent of the volume, so that for any disjoint collection $\\big\\{B_{r_i}(x_i):x_i\\in \\mathcal S_{\\epsilon,\\,\\beta r_i}^k(X)\\cap B_1, \\,r_i\\le 1\\big\\}$, the packing estimate $\\sum r_i^k\\le C$ holds. Consequently, we obt"},"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":"1912.03615","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2019-12-08T04:51:47Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"52c7f7dcf9a4890ca9f33d602eddb8b026f1a5a5992540523423c797905c131a","abstract_canon_sha256":"78c7936f76e365f3e1b3f485e9442f759e52659b79c4fe5b81be5dcfd8ea3a4b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:24:47.117978Z","signature_b64":"oc4S8Pfy3thqu5udTS/nG/crCjbzKBIwG8WU6hirdUTm1S1++Bq81O/+VKwA27Z3lGjU0IuQmkZ9yWNAfO1FBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92dae433b6b66c0d60324e479e4c74e090a9313856bdef91442c8d89b6f32818","last_reissued_at":"2026-07-05T00:24:47.117466Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:24:47.117466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quantitative Estimates on the Singular Sets of Alexandrov Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Aaron Naber, Nan Li","submitted_at":"2019-12-08T04:51:47Z","abstract_excerpt":"Let $X\\in\\text{Alex}\\,^n(-1)$ be an $n$-dimensional Alexandrov space with curvature $\\ge -1$. Let the $r$-scale $(k,\\epsilon)$-singular set $\\mathcal S^k_{\\epsilon,\\,r}(X)$ be the collection of $x\\in X$ so that $B_r(x)$ is not $\\epsilon r$-close to a ball in any splitting space $\\mathbb R^{k+1}\\times Z$. We show that there exists $C(n,\\epsilon)>0$ and $\\beta(n,\\epsilon)>0$, independent of the volume, so that for any disjoint collection $\\big\\{B_{r_i}(x_i):x_i\\in \\mathcal S_{\\epsilon,\\,\\beta r_i}^k(X)\\cap B_1, \\,r_i\\le 1\\big\\}$, the packing estimate $\\sum r_i^k\\le C$ holds. Consequently, we obt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.03615","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1912.03615/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1912.03615","created_at":"2026-07-05T00:24:47.117526+00:00"},{"alias_kind":"arxiv_version","alias_value":"1912.03615v1","created_at":"2026-07-05T00:24:47.117526+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1912.03615","created_at":"2026-07-05T00:24:47.117526+00:00"},{"alias_kind":"pith_short_12","alias_value":"SLNOIM5WWZWA","created_at":"2026-07-05T00:24:47.117526+00:00"},{"alias_kind":"pith_short_16","alias_value":"SLNOIM5WWZWA2YBS","created_at":"2026-07-05T00:24:47.117526+00:00"},{"alias_kind":"pith_short_8","alias_value":"SLNOIM5W","created_at":"2026-07-05T00:24:47.117526+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/SLNOIM5WWZWA2YBSJZDZ4TDU4C","json":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C.json","graph_json":"https://pith.science/api/pith-number/SLNOIM5WWZWA2YBSJZDZ4TDU4C/graph.json","events_json":"https://pith.science/api/pith-number/SLNOIM5WWZWA2YBSJZDZ4TDU4C/events.json","paper":"https://pith.science/paper/SLNOIM5W"},"agent_actions":{"view_html":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C","download_json":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C.json","view_paper":"https://pith.science/paper/SLNOIM5W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1912.03615&json=true","fetch_graph":"https://pith.science/api/pith-number/SLNOIM5WWZWA2YBSJZDZ4TDU4C/graph.json","fetch_events":"https://pith.science/api/pith-number/SLNOIM5WWZWA2YBSJZDZ4TDU4C/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/action/storage_attestation","attest_author":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/action/author_attestation","sign_citation":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/action/citation_signature","submit_replication":"https://pith.science/pith/SLNOIM5WWZWA2YBSJZDZ4TDU4C/action/replication_record"}},"created_at":"2026-07-05T00:24:47.117526+00:00","updated_at":"2026-07-05T00:24:47.117526+00:00"}