{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:MIQ56OL7B3QH2Q4KNMAER24TI5","short_pith_number":"pith:MIQ56OL7","schema_version":"1.0","canonical_sha256":"6221df397f0ee07d438a6b0048eb9347471d11a7ace449ed884a934abf53b143","source":{"kind":"arxiv","id":"1808.08465","version":4},"attestation_state":"computed","paper":{"title":"Stronger sum-product inequalities for small sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"George Shakan, Ilya Shkredov, Misha Rudnev","submitted_at":"2018-08-25T20:03:17Z","abstract_excerpt":"Let $F$ be a field and a finite $A\\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$.\n  We strengthen the \"threshold\" sum-product inequality $$|AA|^3 |A\\pm A|^2 \\gg |A|^6\\,,\\;\\;\\;\\;\\mbox{hence} \\;\\; \\;\\;|AA|+|A+A|\\gg |A|^{1+\\frac{1}{5}},$$ due to Roche-Newton, Rudnev and Shkredov, to\n  $$|AA|^5 |A\\pm A|^4 \\gg |A|^{11-o(1)}\\,,\\;\\;\\;\\;\\mbox{hence} \\;\\; \\;\\;|AA|+|A\\pm A|\\gg |A|^{1+\\frac{2}{9}-o(1)},$$ as well as $$ |AA|^{36}|A-A|^{24} \\gg |A|^{73-o(1)}. $$ The latter inequality is \"threshold-breaking\", for it shows for $\\epsilon>0$, one has $$|AA| \\le |A|^{1+\\epsi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.08465","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-25T20:03:17Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e0d32098e0033af8d3906de38f4e0b55d789a0b391ac3593c062fec61dcb48d8","abstract_canon_sha256":"236dac35f0c7b9f303c69078408726d947a4a88899c8a2af71652b746fa88e08"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:44.472830Z","signature_b64":"Dr5VEyPB1mzRkczpIm80zsMATkD0OXFMISNd6+y5qUEhN31w0/5Ey8eU3O6key89ZQuXBwpalPXLioxEGMAzDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6221df397f0ee07d438a6b0048eb9347471d11a7ace449ed884a934abf53b143","last_reissued_at":"2026-05-18T00:04:44.472021Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:44.472021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stronger sum-product inequalities for small sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"George Shakan, Ilya Shkredov, Misha Rudnev","submitted_at":"2018-08-25T20:03:17Z","abstract_excerpt":"Let $F$ be a field and a finite $A\\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$.\n  We strengthen the \"threshold\" sum-product inequality $$|AA|^3 |A\\pm A|^2 \\gg |A|^6\\,,\\;\\;\\;\\;\\mbox{hence} \\;\\; \\;\\;|AA|+|A+A|\\gg |A|^{1+\\frac{1}{5}},$$ due to Roche-Newton, Rudnev and Shkredov, to\n  $$|AA|^5 |A\\pm A|^4 \\gg |A|^{11-o(1)}\\,,\\;\\;\\;\\;\\mbox{hence} \\;\\; \\;\\;|AA|+|A\\pm A|\\gg |A|^{1+\\frac{2}{9}-o(1)},$$ as well as $$ |AA|^{36}|A-A|^{24} \\gg |A|^{73-o(1)}. $$ The latter inequality is \"threshold-breaking\", for it shows for $\\epsilon>0$, one has $$|AA| \\le |A|^{1+\\epsi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08465","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":"1808.08465","created_at":"2026-05-18T00:04:44.472168+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.08465v4","created_at":"2026-05-18T00:04:44.472168+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.08465","created_at":"2026-05-18T00:04:44.472168+00:00"},{"alias_kind":"pith_short_12","alias_value":"MIQ56OL7B3QH","created_at":"2026-05-18T12:32:37.024351+00:00"},{"alias_kind":"pith_short_16","alias_value":"MIQ56OL7B3QH2Q4K","created_at":"2026-05-18T12:32:37.024351+00:00"},{"alias_kind":"pith_short_8","alias_value":"MIQ56OL7","created_at":"2026-05-18T12:32:37.024351+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1907.03357","citing_title":"Some remarks on products of sets in the Heisenberg group and in the affine group","ref_index":14,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MIQ56OL7B3QH2Q4KNMAER24TI5","json":"https://pith.science/pith/MIQ56OL7B3QH2Q4KNMAER24TI5.json","graph_json":"https://pith.science/api/pith-number/MIQ56OL7B3QH2Q4KNMAER24TI5/graph.json","events_json":"https://pith.science/api/pith-number/MIQ56OL7B3QH2Q4KNMAER24TI5/events.json","paper":"https://pith.science/paper/MIQ56OL7"},"agent_actions":{"view_html":"https://pith.science/pith/MIQ56OL7B3QH2Q4KNMAER24TI5","download_json":"https://pith.science/pith/MIQ56OL7B3QH2Q4KNMAER24TI5.json","view_paper":"https://pith.science/paper/MIQ56OL7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.08465&json=true","fetch_graph":"https://pith.science/api/pith-number/MIQ56OL7B3QH2Q4KNMAER24TI5/graph.json","fetch_events":"https://pith.science/api/pith-number/MIQ56OL7B3QH2Q4KNMAER24TI5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MIQ56OL7B3QH2Q4KNMAER24TI5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MIQ56OL7B3QH2Q4KNMAER24TI5/action/storage_attestation","attest_author":"https://pith.science/pith/MIQ56OL7B3QH2Q4KNMAER24TI5/action/author_attestation","sign_citation":"https://pith.science/pith/MIQ56OL7B3QH2Q4KNMAER24TI5/action/citation_signature","submit_replication":"https://pith.science/pith/MIQ56OL7B3QH2Q4KNMAER24TI5/action/replication_record"}},"created_at":"2026-05-18T00:04:44.472168+00:00","updated_at":"2026-05-18T00:04:44.472168+00:00"}