{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:ORDRLBRRSIOTZMA6VOQIN55XCZ","short_pith_number":"pith:ORDRLBRR","schema_version":"1.0","canonical_sha256":"7447158631921d3cb01eaba086f7b7166bf48ea132f5404f9165ea502aff4f8a","source":{"kind":"arxiv","id":"1409.7349","version":4},"attestation_state":"computed","paper":{"title":"Few products, many h-fold sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Albert Bush, Ernie Croot","submitted_at":"2014-09-25T17:53:42Z","abstract_excerpt":"Improving upon a technique of Croot and Hart, we show that for every $h$, there exists an $\\epsilon > 0$ such that if $A \\subseteq \\mathbb{R}$ is sufficiently large and $|A.A| \\le |A|^{1+\\epsilon}$, then $|hA| \\ge |A|^{\\Omega(e^{\\sqrt{c\\log{h}}})}$."},"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":"1409.7349","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-25T17:53:42Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"1eb8ad8d3877bd9130b4e8e4e7a4f32ecf5fc78f4df9c5816558bd44506fba44","abstract_canon_sha256":"c68b6ff546bf2138a6f5b1ae7756af99013e6f19f01190d2240d4f7f74282d6c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:47.805216Z","signature_b64":"Z8dhvAK7f8WDS5MWLEB+/AsqayA+JI7R12gIEOmAUNfYEACFlz5g9n8/DU8IUfwtB4EGE4fz8TccFxB3MhPQDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7447158631921d3cb01eaba086f7b7166bf48ea132f5404f9165ea502aff4f8a","last_reissued_at":"2026-05-18T02:39:47.804846Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:47.804846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Few products, many h-fold sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Albert Bush, Ernie Croot","submitted_at":"2014-09-25T17:53:42Z","abstract_excerpt":"Improving upon a technique of Croot and Hart, we show that for every $h$, there exists an $\\epsilon > 0$ such that if $A \\subseteq \\mathbb{R}$ is sufficiently large and $|A.A| \\le |A|^{1+\\epsilon}$, then $|hA| \\ge |A|^{\\Omega(e^{\\sqrt{c\\log{h}}})}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7349","kind":"arxiv","version":4},"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":"1409.7349","created_at":"2026-05-18T02:39:47.804902+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.7349v4","created_at":"2026-05-18T02:39:47.804902+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.7349","created_at":"2026-05-18T02:39:47.804902+00:00"},{"alias_kind":"pith_short_12","alias_value":"ORDRLBRRSIOT","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_16","alias_value":"ORDRLBRRSIOTZMA6","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_8","alias_value":"ORDRLBRR","created_at":"2026-05-18T12:28:43.426989+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/ORDRLBRRSIOTZMA6VOQIN55XCZ","json":"https://pith.science/pith/ORDRLBRRSIOTZMA6VOQIN55XCZ.json","graph_json":"https://pith.science/api/pith-number/ORDRLBRRSIOTZMA6VOQIN55XCZ/graph.json","events_json":"https://pith.science/api/pith-number/ORDRLBRRSIOTZMA6VOQIN55XCZ/events.json","paper":"https://pith.science/paper/ORDRLBRR"},"agent_actions":{"view_html":"https://pith.science/pith/ORDRLBRRSIOTZMA6VOQIN55XCZ","download_json":"https://pith.science/pith/ORDRLBRRSIOTZMA6VOQIN55XCZ.json","view_paper":"https://pith.science/paper/ORDRLBRR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.7349&json=true","fetch_graph":"https://pith.science/api/pith-number/ORDRLBRRSIOTZMA6VOQIN55XCZ/graph.json","fetch_events":"https://pith.science/api/pith-number/ORDRLBRRSIOTZMA6VOQIN55XCZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ORDRLBRRSIOTZMA6VOQIN55XCZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ORDRLBRRSIOTZMA6VOQIN55XCZ/action/storage_attestation","attest_author":"https://pith.science/pith/ORDRLBRRSIOTZMA6VOQIN55XCZ/action/author_attestation","sign_citation":"https://pith.science/pith/ORDRLBRRSIOTZMA6VOQIN55XCZ/action/citation_signature","submit_replication":"https://pith.science/pith/ORDRLBRRSIOTZMA6VOQIN55XCZ/action/replication_record"}},"created_at":"2026-05-18T02:39:47.804902+00:00","updated_at":"2026-05-18T02:39:47.804902+00:00"}