{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:JMMSZ5SARR7RDBCIB5ZQGJW3TD","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":"18931dba2e574cb51644e5e0ca7a4f371a8e15d3a4cb668cc40c082b87c92255","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-29T12:00:49Z","title_canon_sha256":"4dd935e293abc34cdeb6882f4daa0b2589e341d2bf8a222ecc0024659f3692d9"},"schema_version":"1.0","source":{"id":"1309.7579","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.7579","created_at":"2026-05-18T03:11:56Z"},{"alias_kind":"arxiv_version","alias_value":"1309.7579v1","created_at":"2026-05-18T03:11:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7579","created_at":"2026-05-18T03:11:56Z"},{"alias_kind":"pith_short_12","alias_value":"JMMSZ5SARR7R","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"JMMSZ5SARR7RDBCI","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"JMMSZ5SA","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:6b4db5fa44926f81a2c2d18556f171705e9db8f7e837ae8bdef9cdd342035b47","target":"graph","created_at":"2026-05-18T03:11:56Z","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":"We show that for a sufficiently big \\textit{brick} $B$ of the $(2n+1)$-dimensional Heisenberg group $H_n$ over the finite field $\\mathbb{F}_p$, the product set $B\\cdot B$ contains at least $|B|/p$ many cosets of some non trivial subgroup of $H_n$.","authors_text":"Fran\\c{c}ois Hennecart, Norbert Hegyv\\'ari","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-29T12:00:49Z","title":"A Structure result for bricks in Heisenberg groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7579","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":"sha256:619b9d13c490d7d4346552ffd8b1b600ac79831be7c4a9448984f0befafe4d4e","target":"record","created_at":"2026-05-18T03:11:56Z","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":"18931dba2e574cb51644e5e0ca7a4f371a8e15d3a4cb668cc40c082b87c92255","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-29T12:00:49Z","title_canon_sha256":"4dd935e293abc34cdeb6882f4daa0b2589e341d2bf8a222ecc0024659f3692d9"},"schema_version":"1.0","source":{"id":"1309.7579","kind":"arxiv","version":1}},"canonical_sha256":"4b192cf6408c7f1184480f730326db98ee2817990ffd83567c4c76675994da4f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b192cf6408c7f1184480f730326db98ee2817990ffd83567c4c76675994da4f","first_computed_at":"2026-05-18T03:11:56.508624Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:11:56.508624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PMxj+IxYLRtCax1Uf/OACUaU7qKazlL/F9m48jX+WsgG9lU8qm38NyCLbevhuxHCa6D5Cmf8SWYElCGMDuDbBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:11:56.509262Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.7579","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:619b9d13c490d7d4346552ffd8b1b600ac79831be7c4a9448984f0befafe4d4e","sha256:6b4db5fa44926f81a2c2d18556f171705e9db8f7e837ae8bdef9cdd342035b47"],"state_sha256":"36f499f8a8488474c5b979624a5f974e729228af936561908aec48a0156175db"}