{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:PQ4ZA5XLCWW3AORVXHYQUPRWHF","short_pith_number":"pith:PQ4ZA5XL","schema_version":"1.0","canonical_sha256":"7c399076eb15adb03a35b9f10a3e3639723666eec5341b9030cc7a09d36be8df","source":{"kind":"arxiv","id":"1202.0472","version":2},"attestation_state":"computed","paper":{"title":"Tangency properties of sets with finite geometric curvature energies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Sebastian Scholtes","submitted_at":"2012-02-02T16:07:54Z","abstract_excerpt":"We investigate inverse thickness $1/\\Delta$ and the integral Menger curvature energies $\\mathcal{U}_{p}^{\\alpha}$, $\\mathcal{I}_{p}^{\\alpha}$ and $\\mathcal{M}_{p}^{\\alpha}$, to find that finite $1/\\Delta$ or $\\mathcal{U}_{p}^{\\alpha}$ implies the existence of an approximate $\\alpha$-tangent at all points of the set, when $p\\geq \\alpha$ and that finite $\\mathcal{I}_{p}^{\\alpha}$ or $\\mathcal{M}_{p}^{\\alpha}$ implies the existence of a weak approximate $\\alpha$-tangent at every point of the set for $p\\geq 2\\alpha$ or $p\\geq 3\\alpha$, respectively, if some additional density properties hold. This"},"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":"1202.0472","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-02-02T16:07:54Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"d183b53e0c727792a78fb97540f29bf40a2ffc279e119346a229321f2b9ff404","abstract_canon_sha256":"5c35592c55d017b3e9af19394343e49d7b959f182a62f4c6bf6e20b0df1a6fbb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:39.366402Z","signature_b64":"OVRYETZTrKzpsH1Lt8dPGWqpy+w0zxeep6lmLz7aP/9KiJBv0Z+5gEoPyr4MZ5s3A/SS9vntsdIU1GCfoa/FDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c399076eb15adb03a35b9f10a3e3639723666eec5341b9030cc7a09d36be8df","last_reissued_at":"2026-05-18T03:58:39.365791Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:39.365791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tangency properties of sets with finite geometric curvature energies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Sebastian Scholtes","submitted_at":"2012-02-02T16:07:54Z","abstract_excerpt":"We investigate inverse thickness $1/\\Delta$ and the integral Menger curvature energies $\\mathcal{U}_{p}^{\\alpha}$, $\\mathcal{I}_{p}^{\\alpha}$ and $\\mathcal{M}_{p}^{\\alpha}$, to find that finite $1/\\Delta$ or $\\mathcal{U}_{p}^{\\alpha}$ implies the existence of an approximate $\\alpha$-tangent at all points of the set, when $p\\geq \\alpha$ and that finite $\\mathcal{I}_{p}^{\\alpha}$ or $\\mathcal{M}_{p}^{\\alpha}$ implies the existence of a weak approximate $\\alpha$-tangent at every point of the set for $p\\geq 2\\alpha$ or $p\\geq 3\\alpha$, respectively, if some additional density properties hold. This"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0472","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":"1202.0472","created_at":"2026-05-18T03:58:39.365895+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.0472v2","created_at":"2026-05-18T03:58:39.365895+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.0472","created_at":"2026-05-18T03:58:39.365895+00:00"},{"alias_kind":"pith_short_12","alias_value":"PQ4ZA5XLCWW3","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"PQ4ZA5XLCWW3AORV","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"PQ4ZA5XL","created_at":"2026-05-18T12:27:18.751474+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/PQ4ZA5XLCWW3AORVXHYQUPRWHF","json":"https://pith.science/pith/PQ4ZA5XLCWW3AORVXHYQUPRWHF.json","graph_json":"https://pith.science/api/pith-number/PQ4ZA5XLCWW3AORVXHYQUPRWHF/graph.json","events_json":"https://pith.science/api/pith-number/PQ4ZA5XLCWW3AORVXHYQUPRWHF/events.json","paper":"https://pith.science/paper/PQ4ZA5XL"},"agent_actions":{"view_html":"https://pith.science/pith/PQ4ZA5XLCWW3AORVXHYQUPRWHF","download_json":"https://pith.science/pith/PQ4ZA5XLCWW3AORVXHYQUPRWHF.json","view_paper":"https://pith.science/paper/PQ4ZA5XL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.0472&json=true","fetch_graph":"https://pith.science/api/pith-number/PQ4ZA5XLCWW3AORVXHYQUPRWHF/graph.json","fetch_events":"https://pith.science/api/pith-number/PQ4ZA5XLCWW3AORVXHYQUPRWHF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PQ4ZA5XLCWW3AORVXHYQUPRWHF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PQ4ZA5XLCWW3AORVXHYQUPRWHF/action/storage_attestation","attest_author":"https://pith.science/pith/PQ4ZA5XLCWW3AORVXHYQUPRWHF/action/author_attestation","sign_citation":"https://pith.science/pith/PQ4ZA5XLCWW3AORVXHYQUPRWHF/action/citation_signature","submit_replication":"https://pith.science/pith/PQ4ZA5XLCWW3AORVXHYQUPRWHF/action/replication_record"}},"created_at":"2026-05-18T03:58:39.365895+00:00","updated_at":"2026-05-18T03:58:39.365895+00:00"}