{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:SCBPC7GQ24IXBYSMGBEXNJBYSK","short_pith_number":"pith:SCBPC7GQ","schema_version":"1.0","canonical_sha256":"9082f17cd0d71170e24c304976a43892b63641d779f63d340bcf4f0a59d9fa28","source":{"kind":"arxiv","id":"1606.03468","version":2},"attestation_state":"computed","paper":{"title":"An improved bound on $(A+A)/(A+A)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Lund","submitted_at":"2016-06-10T20:28:21Z","abstract_excerpt":"We show that, for a finite set $A$ of real numbers, the size of the set $$\\frac{A+A}{A+A} = \\left\\{ \\frac{a+b}{c+d} : a,b,c,d \\in A, c+d \\neq 0 \\right \\}$$ is bounded from below by $$\\left|\\frac{A+A}{A+A} \\right| \\gg \\frac{|A|^{2+1/4}}{|A / A|^{1/8} \\log |A|}.$$ This improves a result of Roche-Newton."},"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":"1606.03468","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-10T20:28:21Z","cross_cats_sorted":[],"title_canon_sha256":"8209d6229263192cfc2cdf4fd3f47f0b2443dc8d5d55264bbab138161fd42c3e","abstract_canon_sha256":"75c77ac74878645d305571f0cacf150353b419bf1f5c2e65056cb3cdb654a19b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:30.540406Z","signature_b64":"IUNs4fLqPvgi4GY8RGoaADHw7XhDqyOG16tkET8uNGCQRP5rdsyydP9XJc3VfdDZv2Dewcypc6stC0ZhSDJ5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9082f17cd0d71170e24c304976a43892b63641d779f63d340bcf4f0a59d9fa28","last_reissued_at":"2026-05-18T01:02:30.539763Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:30.539763Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An improved bound on $(A+A)/(A+A)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Lund","submitted_at":"2016-06-10T20:28:21Z","abstract_excerpt":"We show that, for a finite set $A$ of real numbers, the size of the set $$\\frac{A+A}{A+A} = \\left\\{ \\frac{a+b}{c+d} : a,b,c,d \\in A, c+d \\neq 0 \\right \\}$$ is bounded from below by $$\\left|\\frac{A+A}{A+A} \\right| \\gg \\frac{|A|^{2+1/4}}{|A / A|^{1/8} \\log |A|}.$$ This improves a result of Roche-Newton."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03468","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":"1606.03468","created_at":"2026-05-18T01:02:30.539852+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.03468v2","created_at":"2026-05-18T01:02:30.539852+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.03468","created_at":"2026-05-18T01:02:30.539852+00:00"},{"alias_kind":"pith_short_12","alias_value":"SCBPC7GQ24IX","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_16","alias_value":"SCBPC7GQ24IXBYSM","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_8","alias_value":"SCBPC7GQ","created_at":"2026-05-18T12:30:44.179134+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/SCBPC7GQ24IXBYSMGBEXNJBYSK","json":"https://pith.science/pith/SCBPC7GQ24IXBYSMGBEXNJBYSK.json","graph_json":"https://pith.science/api/pith-number/SCBPC7GQ24IXBYSMGBEXNJBYSK/graph.json","events_json":"https://pith.science/api/pith-number/SCBPC7GQ24IXBYSMGBEXNJBYSK/events.json","paper":"https://pith.science/paper/SCBPC7GQ"},"agent_actions":{"view_html":"https://pith.science/pith/SCBPC7GQ24IXBYSMGBEXNJBYSK","download_json":"https://pith.science/pith/SCBPC7GQ24IXBYSMGBEXNJBYSK.json","view_paper":"https://pith.science/paper/SCBPC7GQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.03468&json=true","fetch_graph":"https://pith.science/api/pith-number/SCBPC7GQ24IXBYSMGBEXNJBYSK/graph.json","fetch_events":"https://pith.science/api/pith-number/SCBPC7GQ24IXBYSMGBEXNJBYSK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SCBPC7GQ24IXBYSMGBEXNJBYSK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SCBPC7GQ24IXBYSMGBEXNJBYSK/action/storage_attestation","attest_author":"https://pith.science/pith/SCBPC7GQ24IXBYSMGBEXNJBYSK/action/author_attestation","sign_citation":"https://pith.science/pith/SCBPC7GQ24IXBYSMGBEXNJBYSK/action/citation_signature","submit_replication":"https://pith.science/pith/SCBPC7GQ24IXBYSMGBEXNJBYSK/action/replication_record"}},"created_at":"2026-05-18T01:02:30.539852+00:00","updated_at":"2026-05-18T01:02:30.539852+00:00"}