{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:KHA54SR3RIRC5EHJTV6ZB6HW4T","short_pith_number":"pith:KHA54SR3","schema_version":"1.0","canonical_sha256":"51c1de4a3b8a222e90e99d7d90f8f6e4ff0e43e029c955e691047e6176e0fd44","source":{"kind":"arxiv","id":"1606.02320","version":3},"attestation_state":"computed","paper":{"title":"On additive bases of sets with small product set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dmitrii Zhelezov, Ilya D. Shkredov","submitted_at":"2016-06-07T20:12:50Z","abstract_excerpt":"We prove that finite sets of real numbers satisfying $|AA| \\leq |A|^{1+\\epsilon}$ with sufficiently small $\\epsilon > 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \\geq 2$. The result can be seen as a real analog of the conjecture of S\\'ark\\\"ozy that multiplicative subgroups of finite fields of prime order are additively irreducible."},"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":"1606.02320","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-07T20:12:50Z","cross_cats_sorted":[],"title_canon_sha256":"21d3cce05246a28c57db82078e2ec2d3e3725a51b3c0d0c71e115bd3fb0cd506","abstract_canon_sha256":"c921b0974e86b58fd22a93ac25a1a91e4d4044a556d1e5cfc4f9ae35dd1e0d02"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:37.811097Z","signature_b64":"ceODgKYbTXvsurvZL/TcCMm3KNy0c8OkcXTt5fkvEgIorM36PAmGAptW79G0PMiGrRWBxdKkhzd8vsqSLlwaCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51c1de4a3b8a222e90e99d7d90f8f6e4ff0e43e029c955e691047e6176e0fd44","last_reissued_at":"2026-05-18T00:57:37.810545Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:37.810545Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On additive bases of sets with small product set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dmitrii Zhelezov, Ilya D. Shkredov","submitted_at":"2016-06-07T20:12:50Z","abstract_excerpt":"We prove that finite sets of real numbers satisfying $|AA| \\leq |A|^{1+\\epsilon}$ with sufficiently small $\\epsilon > 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \\geq 2$. The result can be seen as a real analog of the conjecture of S\\'ark\\\"ozy that multiplicative subgroups of finite fields of prime order are additively irreducible."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02320","kind":"arxiv","version":3},"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":"1606.02320","created_at":"2026-05-18T00:57:37.810646+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.02320v3","created_at":"2026-05-18T00:57:37.810646+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02320","created_at":"2026-05-18T00:57:37.810646+00:00"},{"alias_kind":"pith_short_12","alias_value":"KHA54SR3RIRC","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"KHA54SR3RIRC5EHJ","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"KHA54SR3","created_at":"2026-05-18T12:30:25.849896+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/KHA54SR3RIRC5EHJTV6ZB6HW4T","json":"https://pith.science/pith/KHA54SR3RIRC5EHJTV6ZB6HW4T.json","graph_json":"https://pith.science/api/pith-number/KHA54SR3RIRC5EHJTV6ZB6HW4T/graph.json","events_json":"https://pith.science/api/pith-number/KHA54SR3RIRC5EHJTV6ZB6HW4T/events.json","paper":"https://pith.science/paper/KHA54SR3"},"agent_actions":{"view_html":"https://pith.science/pith/KHA54SR3RIRC5EHJTV6ZB6HW4T","download_json":"https://pith.science/pith/KHA54SR3RIRC5EHJTV6ZB6HW4T.json","view_paper":"https://pith.science/paper/KHA54SR3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.02320&json=true","fetch_graph":"https://pith.science/api/pith-number/KHA54SR3RIRC5EHJTV6ZB6HW4T/graph.json","fetch_events":"https://pith.science/api/pith-number/KHA54SR3RIRC5EHJTV6ZB6HW4T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KHA54SR3RIRC5EHJTV6ZB6HW4T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KHA54SR3RIRC5EHJTV6ZB6HW4T/action/storage_attestation","attest_author":"https://pith.science/pith/KHA54SR3RIRC5EHJTV6ZB6HW4T/action/author_attestation","sign_citation":"https://pith.science/pith/KHA54SR3RIRC5EHJTV6ZB6HW4T/action/citation_signature","submit_replication":"https://pith.science/pith/KHA54SR3RIRC5EHJTV6ZB6HW4T/action/replication_record"}},"created_at":"2026-05-18T00:57:37.810646+00:00","updated_at":"2026-05-18T00:57:37.810646+00:00"}