{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:KNUI7UDXTHKRDHSOOC5DISNAT4","short_pith_number":"pith:KNUI7UDX","schema_version":"1.0","canonical_sha256":"53688fd07799d5119e4e70ba3449a09f24211ee47b9097b875f70a27c4d941a5","source":{"kind":"arxiv","id":"1002.1605","version":2},"attestation_state":"computed","paper":{"title":"Growth of small generating sets in SL_n(Z/pZ)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.GR","authors_text":"Harald Andres Helfgott, Nick Gill","submitted_at":"2010-02-08T15:48:28Z","abstract_excerpt":"Let G=SL_n. Let K=Z/pZ, p a prime. Let A\\subset G(K) generate G(K). Suppose that |A|<p^{n+1-\\delta}, delta>0. Then |A A A|>>|A|^{1+\\epsilon}, where epsilon>0 and the implied constant depend only on n and delta."},"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":"1002.1605","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-08T15:48:28Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"0db69f00b4a929cc98c18d3be661044006486e6846d3e6d9646f5f4804489c52","abstract_canon_sha256":"87843fffedb993259d670ac0a2751f8d2ae3fa907ea46ac0d2389cb3d40d5585"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:05.918807Z","signature_b64":"cXsXvS0xQROYAHwnXuJdsDYqJRDjGpAw0X5nqQY+E6ZnH60GVNio/Lz5ZJf6XIvKcyqtk/3KpSCyLvJ49P0EAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53688fd07799d5119e4e70ba3449a09f24211ee47b9097b875f70a27c4d941a5","last_reissued_at":"2026-05-18T04:41:05.918440Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:05.918440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Growth of small generating sets in SL_n(Z/pZ)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.GR","authors_text":"Harald Andres Helfgott, Nick Gill","submitted_at":"2010-02-08T15:48:28Z","abstract_excerpt":"Let G=SL_n. Let K=Z/pZ, p a prime. Let A\\subset G(K) generate G(K). Suppose that |A|<p^{n+1-\\delta}, delta>0. Then |A A A|>>|A|^{1+\\epsilon}, where epsilon>0 and the implied constant depend only on n and delta."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.1605","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":"1002.1605","created_at":"2026-05-18T04:41:05.918502+00:00"},{"alias_kind":"arxiv_version","alias_value":"1002.1605v2","created_at":"2026-05-18T04:41:05.918502+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.1605","created_at":"2026-05-18T04:41:05.918502+00:00"},{"alias_kind":"pith_short_12","alias_value":"KNUI7UDXTHKR","created_at":"2026-05-18T12:26:09.077623+00:00"},{"alias_kind":"pith_short_16","alias_value":"KNUI7UDXTHKRDHSO","created_at":"2026-05-18T12:26:09.077623+00:00"},{"alias_kind":"pith_short_8","alias_value":"KNUI7UDX","created_at":"2026-05-18T12:26:09.077623+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/KNUI7UDXTHKRDHSOOC5DISNAT4","json":"https://pith.science/pith/KNUI7UDXTHKRDHSOOC5DISNAT4.json","graph_json":"https://pith.science/api/pith-number/KNUI7UDXTHKRDHSOOC5DISNAT4/graph.json","events_json":"https://pith.science/api/pith-number/KNUI7UDXTHKRDHSOOC5DISNAT4/events.json","paper":"https://pith.science/paper/KNUI7UDX"},"agent_actions":{"view_html":"https://pith.science/pith/KNUI7UDXTHKRDHSOOC5DISNAT4","download_json":"https://pith.science/pith/KNUI7UDXTHKRDHSOOC5DISNAT4.json","view_paper":"https://pith.science/paper/KNUI7UDX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1002.1605&json=true","fetch_graph":"https://pith.science/api/pith-number/KNUI7UDXTHKRDHSOOC5DISNAT4/graph.json","fetch_events":"https://pith.science/api/pith-number/KNUI7UDXTHKRDHSOOC5DISNAT4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KNUI7UDXTHKRDHSOOC5DISNAT4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KNUI7UDXTHKRDHSOOC5DISNAT4/action/storage_attestation","attest_author":"https://pith.science/pith/KNUI7UDXTHKRDHSOOC5DISNAT4/action/author_attestation","sign_citation":"https://pith.science/pith/KNUI7UDXTHKRDHSOOC5DISNAT4/action/citation_signature","submit_replication":"https://pith.science/pith/KNUI7UDXTHKRDHSOOC5DISNAT4/action/replication_record"}},"created_at":"2026-05-18T04:41:05.918502+00:00","updated_at":"2026-05-18T04:41:05.918502+00:00"}