{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:4P3G74U5DS74G6VVL4W7NLUDIL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d7a534a518ab60ac486e17f6ac26ade6a75cd3b9e8b41987c36090bfee9a603f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-12T17:12:06Z","title_canon_sha256":"9ae24913ee7caf05ddad9e5793007d39bdd386f142d38fc23361701b7e4aff38"},"schema_version":"1.0","source":{"id":"1412.4058","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.4058","created_at":"2026-05-18T02:31:27Z"},{"alias_kind":"arxiv_version","alias_value":"1412.4058v1","created_at":"2026-05-18T02:31:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.4058","created_at":"2026-05-18T02:31:27Z"},{"alias_kind":"pith_short_12","alias_value":"4P3G74U5DS74","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"4P3G74U5DS74G6VV","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"4P3G74U5","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:e4f31dbe8f2ff8e47bb08344a7af7d75cc7a41204a1f2bcd81935ca1b37bf8cf","target":"graph","created_at":"2026-05-18T02:31:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Bela Bajnok","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-12T17:12:06Z","title":"The $h$-critical number of finite abelian groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4058","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b160a78c5b4a856ae0cd84b231f8149ae12c6f7331780b4ff0c0e716853bcb89","target":"record","created_at":"2026-05-18T02:31:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d7a534a518ab60ac486e17f6ac26ade6a75cd3b9e8b41987c36090bfee9a603f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-12-12T17:12:06Z","title_canon_sha256":"9ae24913ee7caf05ddad9e5793007d39bdd386f142d38fc23361701b7e4aff38"},"schema_version":"1.0","source":{"id":"1412.4058","kind":"arxiv","version":1}},"canonical_sha256":"e3f66ff29d1cbfc37ab55f2df6ae8342e968f29fa0e41ac4a23bfa12a3a75237","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3f66ff29d1cbfc37ab55f2df6ae8342e968f29fa0e41ac4a23bfa12a3a75237","first_computed_at":"2026-05-18T02:31:27.998522Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:31:27.998522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+yTbu+biEbqHs/jdbg1uol4WPoVL6OCdxdHhpJJzQC8vfKK5DZeQRyKx2Enw5K7cX79MHEhUOjheRIrxP2NLDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:31:27.999127Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.4058","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b160a78c5b4a856ae0cd84b231f8149ae12c6f7331780b4ff0c0e716853bcb89","sha256:e4f31dbe8f2ff8e47bb08344a7af7d75cc7a41204a1f2bcd81935ca1b37bf8cf"],"state_sha256":"592796d5c640cb0464a383d29d15981160f6689bf7d715209e9f6698915821af"}