{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:VGZGE4K3UJIVI34WZDDB3FVKAB","short_pith_number":"pith:VGZGE4K3","schema_version":"1.0","canonical_sha256":"a9b262715ba251546f96c8c61d96aa006c0519b0d29d2248a460a573d657ad52","source":{"kind":"arxiv","id":"1801.10431","version":1},"attestation_state":"computed","paper":{"title":"On the size of the set $AA+A$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chun-Yen Shen, Ilya D. Shkredov, Imre Z. Ruzsa, Oliver Roche-Newton","submitted_at":"2018-01-31T13:00:32Z","abstract_excerpt":"It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \\gg |A|^{\\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \\subset \\mathbb R$ such that $|AA+A|=o(|A|^2)$, disproving a conjecture of Balog."},"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":"1801.10431","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-01-31T13:00:32Z","cross_cats_sorted":[],"title_canon_sha256":"b820f72da523d400720b9956b88241efab9e8ee88bf2a468884b72696c36339f","abstract_canon_sha256":"19c96f0eb86a803dd5def7a15cbe777513768ec8d07932b1488a7f9c2981ae75"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:17.646621Z","signature_b64":"rdNfQ11s3r/my+2kn6BzVdoOqCv33TFtXvBWx9GsnJuhdE6KF/ShNpS8nKqWhWGJwe/9EZrtuz3g1shEhQDwCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9b262715ba251546f96c8c61d96aa006c0519b0d29d2248a460a573d657ad52","last_reissued_at":"2026-05-18T00:04:17.645830Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:17.645830Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the size of the set $AA+A$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chun-Yen Shen, Ilya D. Shkredov, Imre Z. Ruzsa, Oliver Roche-Newton","submitted_at":"2018-01-31T13:00:32Z","abstract_excerpt":"It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \\gg |A|^{\\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \\subset \\mathbb R$ such that $|AA+A|=o(|A|^2)$, disproving a conjecture of Balog."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.10431","kind":"arxiv","version":1},"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":"1801.10431","created_at":"2026-05-18T00:04:17.645958+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.10431v1","created_at":"2026-05-18T00:04:17.645958+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.10431","created_at":"2026-05-18T00:04:17.645958+00:00"},{"alias_kind":"pith_short_12","alias_value":"VGZGE4K3UJIV","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_16","alias_value":"VGZGE4K3UJIVI34W","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_8","alias_value":"VGZGE4K3","created_at":"2026-05-18T12:32:59.047623+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/VGZGE4K3UJIVI34WZDDB3FVKAB","json":"https://pith.science/pith/VGZGE4K3UJIVI34WZDDB3FVKAB.json","graph_json":"https://pith.science/api/pith-number/VGZGE4K3UJIVI34WZDDB3FVKAB/graph.json","events_json":"https://pith.science/api/pith-number/VGZGE4K3UJIVI34WZDDB3FVKAB/events.json","paper":"https://pith.science/paper/VGZGE4K3"},"agent_actions":{"view_html":"https://pith.science/pith/VGZGE4K3UJIVI34WZDDB3FVKAB","download_json":"https://pith.science/pith/VGZGE4K3UJIVI34WZDDB3FVKAB.json","view_paper":"https://pith.science/paper/VGZGE4K3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.10431&json=true","fetch_graph":"https://pith.science/api/pith-number/VGZGE4K3UJIVI34WZDDB3FVKAB/graph.json","fetch_events":"https://pith.science/api/pith-number/VGZGE4K3UJIVI34WZDDB3FVKAB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VGZGE4K3UJIVI34WZDDB3FVKAB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VGZGE4K3UJIVI34WZDDB3FVKAB/action/storage_attestation","attest_author":"https://pith.science/pith/VGZGE4K3UJIVI34WZDDB3FVKAB/action/author_attestation","sign_citation":"https://pith.science/pith/VGZGE4K3UJIVI34WZDDB3FVKAB/action/citation_signature","submit_replication":"https://pith.science/pith/VGZGE4K3UJIVI34WZDDB3FVKAB/action/replication_record"}},"created_at":"2026-05-18T00:04:17.645958+00:00","updated_at":"2026-05-18T00:04:17.645958+00:00"}