{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:LT4CDGNM4DBYPZ4LA2QBN7IAST","short_pith_number":"pith:LT4CDGNM","schema_version":"1.0","canonical_sha256":"5cf82199ace0c387e78b06a016fd0094c0756ff92b890c684edb64cc651278b3","source":{"kind":"arxiv","id":"1410.1372","version":2},"attestation_state":"computed","paper":{"title":"Two-Fold Circle-Covering of the Plane under Congruent Voronoi Polygon Conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Jingchao Chen","submitted_at":"2014-10-03T13:43:54Z","abstract_excerpt":"The $k$-coverage problem is to find the minimum number of disks such that each point in a given plane is covered by at least $k$ disks. Under unit disk condition, when $k$=1, this problem has been solved by Kershner in 1939. However, when $k > 1$, it becomes extremely difficult. One tried to tackle this problem with different restrictions. In this paper, we restrict ourself to congruent Voronoi polygon, and prove the minimum density of the two-coverage with such a restriction. Our proof is simpler and more rigorous than that given recently by Yun et al."},"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":"1410.1372","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2014-10-03T13:43:54Z","cross_cats_sorted":[],"title_canon_sha256":"1c08e69df4f30fd466f0e3531a9845b21aaf8dd1eb6b064c09bf6f4a128a7471","abstract_canon_sha256":"64fd509b12ba3c261010a386e0e69aee33dfebac0333353ad5ce515282ddd4b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:37.675632Z","signature_b64":"BlozcAqNM2FOd3aH3cYKtp6fNzVqIl9kPS/tIan81i+b/q3fmYQORDRqX57WGgmDYh4gaKU0IJzPMOzKT8xPAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5cf82199ace0c387e78b06a016fd0094c0756ff92b890c684edb64cc651278b3","last_reissued_at":"2026-05-18T01:16:37.674852Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:37.674852Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two-Fold Circle-Covering of the Plane under Congruent Voronoi Polygon Conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Jingchao Chen","submitted_at":"2014-10-03T13:43:54Z","abstract_excerpt":"The $k$-coverage problem is to find the minimum number of disks such that each point in a given plane is covered by at least $k$ disks. Under unit disk condition, when $k$=1, this problem has been solved by Kershner in 1939. However, when $k > 1$, it becomes extremely difficult. One tried to tackle this problem with different restrictions. In this paper, we restrict ourself to congruent Voronoi polygon, and prove the minimum density of the two-coverage with such a restriction. Our proof is simpler and more rigorous than that given recently by Yun et al."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1372","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":"1410.1372","created_at":"2026-05-18T01:16:37.674979+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.1372v2","created_at":"2026-05-18T01:16:37.674979+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.1372","created_at":"2026-05-18T01:16:37.674979+00:00"},{"alias_kind":"pith_short_12","alias_value":"LT4CDGNM4DBY","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_16","alias_value":"LT4CDGNM4DBYPZ4L","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_8","alias_value":"LT4CDGNM","created_at":"2026-05-18T12:28:38.356838+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/LT4CDGNM4DBYPZ4LA2QBN7IAST","json":"https://pith.science/pith/LT4CDGNM4DBYPZ4LA2QBN7IAST.json","graph_json":"https://pith.science/api/pith-number/LT4CDGNM4DBYPZ4LA2QBN7IAST/graph.json","events_json":"https://pith.science/api/pith-number/LT4CDGNM4DBYPZ4LA2QBN7IAST/events.json","paper":"https://pith.science/paper/LT4CDGNM"},"agent_actions":{"view_html":"https://pith.science/pith/LT4CDGNM4DBYPZ4LA2QBN7IAST","download_json":"https://pith.science/pith/LT4CDGNM4DBYPZ4LA2QBN7IAST.json","view_paper":"https://pith.science/paper/LT4CDGNM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.1372&json=true","fetch_graph":"https://pith.science/api/pith-number/LT4CDGNM4DBYPZ4LA2QBN7IAST/graph.json","fetch_events":"https://pith.science/api/pith-number/LT4CDGNM4DBYPZ4LA2QBN7IAST/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LT4CDGNM4DBYPZ4LA2QBN7IAST/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LT4CDGNM4DBYPZ4LA2QBN7IAST/action/storage_attestation","attest_author":"https://pith.science/pith/LT4CDGNM4DBYPZ4LA2QBN7IAST/action/author_attestation","sign_citation":"https://pith.science/pith/LT4CDGNM4DBYPZ4LA2QBN7IAST/action/citation_signature","submit_replication":"https://pith.science/pith/LT4CDGNM4DBYPZ4LA2QBN7IAST/action/replication_record"}},"created_at":"2026-05-18T01:16:37.674979+00:00","updated_at":"2026-05-18T01:16:37.674979+00:00"}