{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:4P3G74U5DS74G6VVL4W7NLUDIL","short_pith_number":"pith:4P3G74U5","schema_version":"1.0","canonical_sha256":"e3f66ff29d1cbfc37ab55f2df6ae8342e968f29fa0e41ac4a23bfa12a3a75237","source":{"kind":"arxiv","id":"1412.4058","version":1},"attestation_state":"computed","paper":{"title":"The $h$-critical number of finite abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bela Bajnok","submitted_at":"2014-12-12T17:12:06Z","abstract_excerpt":"For a finite abelian group $G$ and a positive integer $h$, the unrestricted (resp.~restricted) $h$-critical number $\\chi(G,h)$ (resp.~$\\chi \\hat{\\;}(G,h)$) of $G$ is defined to be the minimum value of $m$, if exists, for which the $h$-fold unrestricted (resp.~restricted) sumset of every $m$-subset of $G$ equals $G$ itself. Here we determine $\\chi(G,h)$ for all $G$ and $h$; and prove several results for $\\chi \\hat{\\;}(G,h)$, including the cases of any $G$ and $h = 2$, any $G$ and large $h$, and any $h$ for the cyclic group $\\mathbb{Z}_n$ of even order. We also provide a lower bound for $\\chi \\h"},"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":"1412.4058","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-12T17:12:06Z","cross_cats_sorted":[],"title_canon_sha256":"9ae24913ee7caf05ddad9e5793007d39bdd386f142d38fc23361701b7e4aff38","abstract_canon_sha256":"d7a534a518ab60ac486e17f6ac26ade6a75cd3b9e8b41987c36090bfee9a603f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:31:27.999127Z","signature_b64":"+yTbu+biEbqHs/jdbg1uol4WPoVL6OCdxdHhpJJzQC8vfKK5DZeQRyKx2Enw5K7cX79MHEhUOjheRIrxP2NLDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e3f66ff29d1cbfc37ab55f2df6ae8342e968f29fa0e41ac4a23bfa12a3a75237","last_reissued_at":"2026-05-18T02:31:27.998522Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:31:27.998522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $h$-critical number of finite abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bela Bajnok","submitted_at":"2014-12-12T17:12:06Z","abstract_excerpt":"For a finite abelian group $G$ and a positive integer $h$, the unrestricted (resp.~restricted) $h$-critical number $\\chi(G,h)$ (resp.~$\\chi \\hat{\\;}(G,h)$) of $G$ is defined to be the minimum value of $m$, if exists, for which the $h$-fold unrestricted (resp.~restricted) sumset of every $m$-subset of $G$ equals $G$ itself. Here we determine $\\chi(G,h)$ for all $G$ and $h$; and prove several results for $\\chi \\hat{\\;}(G,h)$, including the cases of any $G$ and $h = 2$, any $G$ and large $h$, and any $h$ for the cyclic group $\\mathbb{Z}_n$ of even order. We also provide a lower bound for $\\chi \\h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4058","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":""},"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":"1412.4058","created_at":"2026-05-18T02:31:27.998613+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.4058v1","created_at":"2026-05-18T02:31:27.998613+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.4058","created_at":"2026-05-18T02:31:27.998613+00:00"},{"alias_kind":"pith_short_12","alias_value":"4P3G74U5DS74","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"4P3G74U5DS74G6VV","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"4P3G74U5","created_at":"2026-05-18T12:28:14.216126+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/4P3G74U5DS74G6VVL4W7NLUDIL","json":"https://pith.science/pith/4P3G74U5DS74G6VVL4W7NLUDIL.json","graph_json":"https://pith.science/api/pith-number/4P3G74U5DS74G6VVL4W7NLUDIL/graph.json","events_json":"https://pith.science/api/pith-number/4P3G74U5DS74G6VVL4W7NLUDIL/events.json","paper":"https://pith.science/paper/4P3G74U5"},"agent_actions":{"view_html":"https://pith.science/pith/4P3G74U5DS74G6VVL4W7NLUDIL","download_json":"https://pith.science/pith/4P3G74U5DS74G6VVL4W7NLUDIL.json","view_paper":"https://pith.science/paper/4P3G74U5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.4058&json=true","fetch_graph":"https://pith.science/api/pith-number/4P3G74U5DS74G6VVL4W7NLUDIL/graph.json","fetch_events":"https://pith.science/api/pith-number/4P3G74U5DS74G6VVL4W7NLUDIL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4P3G74U5DS74G6VVL4W7NLUDIL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4P3G74U5DS74G6VVL4W7NLUDIL/action/storage_attestation","attest_author":"https://pith.science/pith/4P3G74U5DS74G6VVL4W7NLUDIL/action/author_attestation","sign_citation":"https://pith.science/pith/4P3G74U5DS74G6VVL4W7NLUDIL/action/citation_signature","submit_replication":"https://pith.science/pith/4P3G74U5DS74G6VVL4W7NLUDIL/action/replication_record"}},"created_at":"2026-05-18T02:31:27.998613+00:00","updated_at":"2026-05-18T02:31:27.998613+00:00"}