{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:UABUM7GIIEXQD76NXY3DVKRGK2","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":"8416f948fda52ae4e6c7845233e63f10e6e4bb071d9f5437ce0aa31e721dd47b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-05-06T08:25:31Z","title_canon_sha256":"4e25996e198301c2676b635c7b2a9c6aeeaf08eeda42dd9916dd617875f6bdaa"},"schema_version":"1.0","source":{"id":"1405.1187","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.1187","created_at":"2026-05-18T00:56:59Z"},{"alias_kind":"arxiv_version","alias_value":"1405.1187v2","created_at":"2026-05-18T00:56:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1187","created_at":"2026-05-18T00:56:59Z"},{"alias_kind":"pith_short_12","alias_value":"UABUM7GIIEXQ","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_16","alias_value":"UABUM7GIIEXQD76N","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_8","alias_value":"UABUM7GI","created_at":"2026-05-18T12:28:52Z"}],"graph_snapshots":[{"event_id":"sha256:84f6106f10e9e3a6e654d0e8af25d7e391daac63ce09cee210183d098f916452","target":"graph","created_at":"2026-05-18T00:56:59Z","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":"Bourgain, Konyagin and Shparlinski obtained a lower bound for the size of the product set AB when A and B are sets of positive rational numbers with numerator and denominator less or equal than Q. We extend and slightly improve that lower bound using a different approach.","authors_text":"Javier Cilleruelo","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-05-06T08:25:31Z","title":"A note on product sets of rationals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1187","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:e79af760c259b1d5dc8119285a9eb0222c8a59550dfc171709f2000130957557","target":"record","created_at":"2026-05-18T00:56:59Z","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":"8416f948fda52ae4e6c7845233e63f10e6e4bb071d9f5437ce0aa31e721dd47b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-05-06T08:25:31Z","title_canon_sha256":"4e25996e198301c2676b635c7b2a9c6aeeaf08eeda42dd9916dd617875f6bdaa"},"schema_version":"1.0","source":{"id":"1405.1187","kind":"arxiv","version":2}},"canonical_sha256":"a003467cc8412f01ffcdbe363aaa26568f1d377e7f03c64b96ba70e17cd21a60","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a003467cc8412f01ffcdbe363aaa26568f1d377e7f03c64b96ba70e17cd21a60","first_computed_at":"2026-05-18T00:56:59.288062Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:59.288062Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VIt7kdawQcD5WBvHDpRhIqFil1JUHS1BJRxYCT7iaEn9mafwqE4yubjQqpyKMhMIvZVlZe1/PnaJoBSy8XnPAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:59.288527Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.1187","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e79af760c259b1d5dc8119285a9eb0222c8a59550dfc171709f2000130957557","sha256:84f6106f10e9e3a6e654d0e8af25d7e391daac63ce09cee210183d098f916452"],"state_sha256":"5d54bf3a6970610b3f7103135cd9031ab1ef870690799c720cff2fb73c86d40d"}