{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:KNUI7UDXTHKRDHSOOC5DISNAT4","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":"87843fffedb993259d670ac0a2751f8d2ae3fa907ea46ac0d2389cb3d40d5585","cross_cats_sorted":["math.CO","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-08T15:48:28Z","title_canon_sha256":"0db69f00b4a929cc98c18d3be661044006486e6846d3e6d9646f5f4804489c52"},"schema_version":"1.0","source":{"id":"1002.1605","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1002.1605","created_at":"2026-05-18T04:41:05Z"},{"alias_kind":"arxiv_version","alias_value":"1002.1605v2","created_at":"2026-05-18T04:41:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.1605","created_at":"2026-05-18T04:41:05Z"},{"alias_kind":"pith_short_12","alias_value":"KNUI7UDXTHKR","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"KNUI7UDXTHKRDHSO","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"KNUI7UDX","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:eb6b0111048a70fc5db1f702a345847c989007f03d53b739300931cc2706ac90","target":"graph","created_at":"2026-05-18T04:41:05Z","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":"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.","authors_text":"Harald Andres Helfgott, Nick Gill","cross_cats":["math.CO","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-08T15:48:28Z","title":"Growth of small generating sets in SL_n(Z/pZ)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.1605","kind":"arxiv","version":2},"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":"sha256:8c1dba7ef5299544a812aec12329c17875af645390d1f5e600d4ca549c8d94f4","target":"record","created_at":"2026-05-18T04:41:05Z","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":"87843fffedb993259d670ac0a2751f8d2ae3fa907ea46ac0d2389cb3d40d5585","cross_cats_sorted":["math.CO","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-08T15:48:28Z","title_canon_sha256":"0db69f00b4a929cc98c18d3be661044006486e6846d3e6d9646f5f4804489c52"},"schema_version":"1.0","source":{"id":"1002.1605","kind":"arxiv","version":2}},"canonical_sha256":"53688fd07799d5119e4e70ba3449a09f24211ee47b9097b875f70a27c4d941a5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"53688fd07799d5119e4e70ba3449a09f24211ee47b9097b875f70a27c4d941a5","first_computed_at":"2026-05-18T04:41:05.918440Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:05.918440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cXsXvS0xQROYAHwnXuJdsDYqJRDjGpAw0X5nqQY+E6ZnH60GVNio/Lz5ZJf6XIvKcyqtk/3KpSCyLvJ49P0EAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:05.918807Z","signed_message":"canonical_sha256_bytes"},"source_id":"1002.1605","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c1dba7ef5299544a812aec12329c17875af645390d1f5e600d4ca549c8d94f4","sha256:eb6b0111048a70fc5db1f702a345847c989007f03d53b739300931cc2706ac90"],"state_sha256":"192dec864f5e5f86322e3a61b6df1126860e8ff1fa9694356cae8de33694fe71"}