{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:Z25C7GDU7MQS3B2576A53MQHHR","short_pith_number":"pith:Z25C7GDU","schema_version":"1.0","canonical_sha256":"ceba2f9874fb212d875dff81ddb2073c6441937bcb79a19aa178f475359ecee1","source":{"kind":"arxiv","id":"1701.01635","version":1},"attestation_state":"computed","paper":{"title":"A Second Wave of Expanders over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brendan Murphy, Giorgis Petridis","submitted_at":"2017-01-06T13:54:47Z","abstract_excerpt":"This is an expository survey on recent sum-product results in finite fields.\n  We present a number of sum-product or \"expander\" results that say that if $|A| > p^{2/3}$ then some set determined by sums and product of elements of $A$ is nearly as large as possible, and if $|A|<p^{2/3}$ then the set in question is significantly larger that $A$. These results are based on a point-plane incidence bound of Rudnev, and are quantitatively stronger than a wave of earlier results following Bourgain, Katz, and Tao's breakthrough sum-product result.\n  In addition, we present two geometric results: an inc"},"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":"1701.01635","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-06T13:54:47Z","cross_cats_sorted":[],"title_canon_sha256":"7b06c35d33be1bce4abf535ef2b79dbd83945cb133b4f32e912409bc3a54b25e","abstract_canon_sha256":"93d7e37b278548b0410d727ae82e0fd88b734ed91c67d1cced586c707416a9de"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:16.350469Z","signature_b64":"wK/rsl+E1LkxRgrHraWib5k0UUBjU9+7zEmKTH+X2fC0LQ2H5t7BTBjKZo+8YVg8O588mn0pj8wYIHtwGq+5Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ceba2f9874fb212d875dff81ddb2073c6441937bcb79a19aa178f475359ecee1","last_reissued_at":"2026-05-18T00:53:16.350057Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:16.350057Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Second Wave of Expanders over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brendan Murphy, Giorgis Petridis","submitted_at":"2017-01-06T13:54:47Z","abstract_excerpt":"This is an expository survey on recent sum-product results in finite fields.\n  We present a number of sum-product or \"expander\" results that say that if $|A| > p^{2/3}$ then some set determined by sums and product of elements of $A$ is nearly as large as possible, and if $|A|<p^{2/3}$ then the set in question is significantly larger that $A$. These results are based on a point-plane incidence bound of Rudnev, and are quantitatively stronger than a wave of earlier results following Bourgain, Katz, and Tao's breakthrough sum-product result.\n  In addition, we present two geometric results: an inc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.01635","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":"1701.01635","created_at":"2026-05-18T00:53:16.350121+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.01635v1","created_at":"2026-05-18T00:53:16.350121+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.01635","created_at":"2026-05-18T00:53:16.350121+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z25C7GDU7MQS","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z25C7GDU7MQS3B25","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z25C7GDU","created_at":"2026-05-18T12:31:59.375834+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/Z25C7GDU7MQS3B2576A53MQHHR","json":"https://pith.science/pith/Z25C7GDU7MQS3B2576A53MQHHR.json","graph_json":"https://pith.science/api/pith-number/Z25C7GDU7MQS3B2576A53MQHHR/graph.json","events_json":"https://pith.science/api/pith-number/Z25C7GDU7MQS3B2576A53MQHHR/events.json","paper":"https://pith.science/paper/Z25C7GDU"},"agent_actions":{"view_html":"https://pith.science/pith/Z25C7GDU7MQS3B2576A53MQHHR","download_json":"https://pith.science/pith/Z25C7GDU7MQS3B2576A53MQHHR.json","view_paper":"https://pith.science/paper/Z25C7GDU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.01635&json=true","fetch_graph":"https://pith.science/api/pith-number/Z25C7GDU7MQS3B2576A53MQHHR/graph.json","fetch_events":"https://pith.science/api/pith-number/Z25C7GDU7MQS3B2576A53MQHHR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z25C7GDU7MQS3B2576A53MQHHR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z25C7GDU7MQS3B2576A53MQHHR/action/storage_attestation","attest_author":"https://pith.science/pith/Z25C7GDU7MQS3B2576A53MQHHR/action/author_attestation","sign_citation":"https://pith.science/pith/Z25C7GDU7MQS3B2576A53MQHHR/action/citation_signature","submit_replication":"https://pith.science/pith/Z25C7GDU7MQS3B2576A53MQHHR/action/replication_record"}},"created_at":"2026-05-18T00:53:16.350121+00:00","updated_at":"2026-05-18T00:53:16.350121+00:00"}