{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:GCPAYRXS4SODK6L3P42ERH5ZTX","short_pith_number":"pith:GCPAYRXS","schema_version":"1.0","canonical_sha256":"309e0c46f2e49c35797b7f34489fb99dd181a6fb0abadaaf218aedf03d85813a","source":{"kind":"arxiv","id":"1712.02823","version":4},"attestation_state":"computed","paper":{"title":"Tangent points of lower content $d$-regular sets and $\\beta$ numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Michele Villa","submitted_at":"2017-12-07T19:26:50Z","abstract_excerpt":"Given a lower content $d$-regular set in $\\mathbb{R}^n$, we prove that the subset of points in $E$ where a certain Dini-type condition on the so-called Jones $\\beta$ numbers holds coincides with the set of tangent points of $E$, up to a set of $\\mathcal{H}^d$-measure zero. The main point of our result is that $\\mathcal{H}^d|_E$ is not assumed to be $\\sigma$-finite; because of this, we use a certain variant of the $\\beta$ coefficient, firstly introduced by Azzam and Schul in [AS1], which is given in terms of integration with respect to the Hausdorff content."},"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":"1712.02823","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-12-07T19:26:50Z","cross_cats_sorted":[],"title_canon_sha256":"28117506c0eccfbd8714ba60b70418a741f7043c2c7311a40012ce8798bd1c9d","abstract_canon_sha256":"40252cd1fd96269fbb7ce21bc961ea0dca4ef81fd9abb62daa0b05a081f68578"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:00:11.959651Z","signature_b64":"wcDRCXdSMaNn6A78iJoqS29ntGnZbDbiV5R4hYSmfka8SERsKGUAgqQmT43zUKWf/cM6HYmMU4iLkwDhfbX/DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"309e0c46f2e49c35797b7f34489fb99dd181a6fb0abadaaf218aedf03d85813a","last_reissued_at":"2026-07-05T00:00:11.959089Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:00:11.959089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tangent points of lower content $d$-regular sets and $\\beta$ numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Michele Villa","submitted_at":"2017-12-07T19:26:50Z","abstract_excerpt":"Given a lower content $d$-regular set in $\\mathbb{R}^n$, we prove that the subset of points in $E$ where a certain Dini-type condition on the so-called Jones $\\beta$ numbers holds coincides with the set of tangent points of $E$, up to a set of $\\mathcal{H}^d$-measure zero. The main point of our result is that $\\mathcal{H}^d|_E$ is not assumed to be $\\sigma$-finite; because of this, we use a certain variant of the $\\beta$ coefficient, firstly introduced by Azzam and Schul in [AS1], which is given in terms of integration with respect to the Hausdorff content."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02823","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1712.02823/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":"1712.02823","created_at":"2026-07-05T00:00:11.959147+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.02823v4","created_at":"2026-07-05T00:00:11.959147+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.02823","created_at":"2026-07-05T00:00:11.959147+00:00"},{"alias_kind":"pith_short_12","alias_value":"GCPAYRXS4SOD","created_at":"2026-07-05T00:00:11.959147+00:00"},{"alias_kind":"pith_short_16","alias_value":"GCPAYRXS4SODK6L3","created_at":"2026-07-05T00:00:11.959147+00:00"},{"alias_kind":"pith_short_8","alias_value":"GCPAYRXS","created_at":"2026-07-05T00:00:11.959147+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.06420","citing_title":"Poincar\\'e Inequalities and Uniform Rectifiability","ref_index":30,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GCPAYRXS4SODK6L3P42ERH5ZTX","json":"https://pith.science/pith/GCPAYRXS4SODK6L3P42ERH5ZTX.json","graph_json":"https://pith.science/api/pith-number/GCPAYRXS4SODK6L3P42ERH5ZTX/graph.json","events_json":"https://pith.science/api/pith-number/GCPAYRXS4SODK6L3P42ERH5ZTX/events.json","paper":"https://pith.science/paper/GCPAYRXS"},"agent_actions":{"view_html":"https://pith.science/pith/GCPAYRXS4SODK6L3P42ERH5ZTX","download_json":"https://pith.science/pith/GCPAYRXS4SODK6L3P42ERH5ZTX.json","view_paper":"https://pith.science/paper/GCPAYRXS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.02823&json=true","fetch_graph":"https://pith.science/api/pith-number/GCPAYRXS4SODK6L3P42ERH5ZTX/graph.json","fetch_events":"https://pith.science/api/pith-number/GCPAYRXS4SODK6L3P42ERH5ZTX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GCPAYRXS4SODK6L3P42ERH5ZTX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GCPAYRXS4SODK6L3P42ERH5ZTX/action/storage_attestation","attest_author":"https://pith.science/pith/GCPAYRXS4SODK6L3P42ERH5ZTX/action/author_attestation","sign_citation":"https://pith.science/pith/GCPAYRXS4SODK6L3P42ERH5ZTX/action/citation_signature","submit_replication":"https://pith.science/pith/GCPAYRXS4SODK6L3P42ERH5ZTX/action/replication_record"}},"created_at":"2026-07-05T00:00:11.959147+00:00","updated_at":"2026-07-05T00:00:11.959147+00:00"}