{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2005:XFUMFODYHK4NYWAABSWWH4PCDM","short_pith_number":"pith:XFUMFODY","schema_version":"1.0","canonical_sha256":"b968c2b8783ab8dc58000cad63f1e21b204de013d0815713cec3ea1b7f7c0c43","source":{"kind":"arxiv","id":"math/0504413","version":2},"attestation_state":"computed","paper":{"title":"A sharp result on m-covers","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Hao Pan, Zhi-Wei Sun","submitted_at":"2005-04-20T14:11:53Z","abstract_excerpt":"Let A={a_s+n_sZ}_{s=1}^k be a finite system of arithmetic sequences which forms an m-cover of Z (i.e., every integer belongs at least to m members of A). In this paper we show the following sharp result: For any positive integers m_1,...,m_k and theta in [0,1), if there is a subset I of {1,...,k} such that the fractional part of sum_{s in I}m_s/n_s is theta, then there are at least 2^m such subsets of {1,...,k}. This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to m-covers of the integral ring of any algebraic number field with a "},"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":"math/0504413","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-04-20T14:11:53Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bf14aa32265090242fb9e55d93869bfac1e972655b6bb4272b58893890fa23f8","abstract_canon_sha256":"6175da51b6df3a792e1b3b2fc720f0a6dd3fc3c01b08fb4178eaf31aaf6cefbc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:50:12.827063Z","signature_b64":"/LRnVDgKXHKddF89Z70furjqO9f+/JwEXE7cbVEIOs7GBP4ZUjyUFuHLOEeWZBVqC0UPBDCRdGxldncUuHGXCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b968c2b8783ab8dc58000cad63f1e21b204de013d0815713cec3ea1b7f7c0c43","last_reissued_at":"2026-07-04T14:50:12.826707Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:50:12.826707Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A sharp result on m-covers","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Hao Pan, Zhi-Wei Sun","submitted_at":"2005-04-20T14:11:53Z","abstract_excerpt":"Let A={a_s+n_sZ}_{s=1}^k be a finite system of arithmetic sequences which forms an m-cover of Z (i.e., every integer belongs at least to m members of A). In this paper we show the following sharp result: For any positive integers m_1,...,m_k and theta in [0,1), if there is a subset I of {1,...,k} such that the fractional part of sum_{s in I}m_s/n_s is theta, then there are at least 2^m such subsets of {1,...,k}. This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to m-covers of the integral ring of any algebraic number field with a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0504413","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0504413/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":"math/0504413","created_at":"2026-07-04T14:50:12.826760+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0504413v2","created_at":"2026-07-04T14:50:12.826760+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0504413","created_at":"2026-07-04T14:50:12.826760+00:00"},{"alias_kind":"pith_short_12","alias_value":"XFUMFODYHK4N","created_at":"2026-07-04T14:50:12.826760+00:00"},{"alias_kind":"pith_short_16","alias_value":"XFUMFODYHK4NYWAA","created_at":"2026-07-04T14:50:12.826760+00:00"},{"alias_kind":"pith_short_8","alias_value":"XFUMFODY","created_at":"2026-07-04T14:50:12.826760+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/XFUMFODYHK4NYWAABSWWH4PCDM","json":"https://pith.science/pith/XFUMFODYHK4NYWAABSWWH4PCDM.json","graph_json":"https://pith.science/api/pith-number/XFUMFODYHK4NYWAABSWWH4PCDM/graph.json","events_json":"https://pith.science/api/pith-number/XFUMFODYHK4NYWAABSWWH4PCDM/events.json","paper":"https://pith.science/paper/XFUMFODY"},"agent_actions":{"view_html":"https://pith.science/pith/XFUMFODYHK4NYWAABSWWH4PCDM","download_json":"https://pith.science/pith/XFUMFODYHK4NYWAABSWWH4PCDM.json","view_paper":"https://pith.science/paper/XFUMFODY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0504413&json=true","fetch_graph":"https://pith.science/api/pith-number/XFUMFODYHK4NYWAABSWWH4PCDM/graph.json","fetch_events":"https://pith.science/api/pith-number/XFUMFODYHK4NYWAABSWWH4PCDM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XFUMFODYHK4NYWAABSWWH4PCDM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XFUMFODYHK4NYWAABSWWH4PCDM/action/storage_attestation","attest_author":"https://pith.science/pith/XFUMFODYHK4NYWAABSWWH4PCDM/action/author_attestation","sign_citation":"https://pith.science/pith/XFUMFODYHK4NYWAABSWWH4PCDM/action/citation_signature","submit_replication":"https://pith.science/pith/XFUMFODYHK4NYWAABSWWH4PCDM/action/replication_record"}},"created_at":"2026-07-04T14:50:12.826760+00:00","updated_at":"2026-07-04T14:50:12.826760+00:00"}