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For \\(E\\subset \\mathbb{F}_q^n\\), define\n  \\[\n  \\Delta_Q(E)=\\{Q(x-y):\\ x,y\\in E\\}.\n  \\]\n  We prove that, for every fixed \\(0<\\alpha<\\frac{1}{2}\\), there exist constants \\(C_\\alpha>0\\) and \\(q_\\alpha\\) such that if \\(q\\ge q_\\alpha\\) and $|E|\\ge C_\\alpha q^{\\frac n2+\\frac13},$\n  then\n  \\[\n  |\\Delta_Q(E)|>1+\\alpha(q-1).\n  \\]\n  In particular, \\(\\Delta_Q(E)\\) contains a"},"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":"2606.29965","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-29T08:43:48Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"4ad60d91937af0e24381018f5cfff1174106e9672e24cedd12bb7851dc32237a","abstract_canon_sha256":"2391ccf8427e6e44537c272af8f44e006a3c35d157199cec41a33144b622d863"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T02:17:42.989376Z","signature_b64":"u/RAmr67xktxxja/0PVX40deZory3AFQjsNNfaZnSmnOu3Y2MMXOXHTaHqnmQfQUZFRYETn4/X2EgZM9v0n9CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"177ce579077bdfa4a7524e6eca9b0115d45b3ffc8a01962ea62db1a7c7e35498","last_reissued_at":"2026-06-30T02:17:42.988860Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T02:17:42.988860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Delsarte Linear Programming Approach to the Erd\\H{o}s--Falconer Distance Problem over Finite Fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CO","authors_text":"Tao Zhang","submitted_at":"2026-06-29T08:43:48Z","abstract_excerpt":"We introduce a Delsarte linear programming approach to the finite field Erd\\H{o}s--Falconer distance problem. Let \\(q\\) be an odd prime power, let \\(n\\) be even, and let \\(Q\\) be a non-degenerate quadratic form on \\(\\mathbb{F}_q^n\\). For \\(E\\subset \\mathbb{F}_q^n\\), define\n  \\[\n  \\Delta_Q(E)=\\{Q(x-y):\\ x,y\\in E\\}.\n  \\]\n  We prove that, for every fixed \\(0<\\alpha<\\frac{1}{2}\\), there exist constants \\(C_\\alpha>0\\) and \\(q_\\alpha\\) such that if \\(q\\ge q_\\alpha\\) and $|E|\\ge C_\\alpha q^{\\frac n2+\\frac13},$\n  then\n  \\[\n  |\\Delta_Q(E)|>1+\\alpha(q-1).\n  \\]\n  In particular, \\(\\Delta_Q(E)\\) contains a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29965","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.29965/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2606.29965","created_at":"2026-06-30T02:17:42.988950+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.29965v1","created_at":"2026-06-30T02:17:42.988950+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.29965","created_at":"2026-06-30T02:17:42.988950+00:00"},{"alias_kind":"pith_short_12","alias_value":"C56OK6IHPPP2","created_at":"2026-06-30T02:17:42.988950+00:00"},{"alias_kind":"pith_short_16","alias_value":"C56OK6IHPPP2JJ2S","created_at":"2026-06-30T02:17:42.988950+00:00"},{"alias_kind":"pith_short_8","alias_value":"C56OK6IH","created_at":"2026-06-30T02:17:42.988950+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/C56OK6IHPPP2JJ2SJZXMVGYBCX","json":"https://pith.science/pith/C56OK6IHPPP2JJ2SJZXMVGYBCX.json","graph_json":"https://pith.science/api/pith-number/C56OK6IHPPP2JJ2SJZXMVGYBCX/graph.json","events_json":"https://pith.science/api/pith-number/C56OK6IHPPP2JJ2SJZXMVGYBCX/events.json","paper":"https://pith.science/paper/C56OK6IH"},"agent_actions":{"view_html":"https://pith.science/pith/C56OK6IHPPP2JJ2SJZXMVGYBCX","download_json":"https://pith.science/pith/C56OK6IHPPP2JJ2SJZXMVGYBCX.json","view_paper":"https://pith.science/paper/C56OK6IH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.29965&json=true","fetch_graph":"https://pith.science/api/pith-number/C56OK6IHPPP2JJ2SJZXMVGYBCX/graph.json","fetch_events":"https://pith.science/api/pith-number/C56OK6IHPPP2JJ2SJZXMVGYBCX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C56OK6IHPPP2JJ2SJZXMVGYBCX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C56OK6IHPPP2JJ2SJZXMVGYBCX/action/storage_attestation","attest_author":"https://pith.science/pith/C56OK6IHPPP2JJ2SJZXMVGYBCX/action/author_attestation","sign_citation":"https://pith.science/pith/C56OK6IHPPP2JJ2SJZXMVGYBCX/action/citation_signature","submit_replication":"https://pith.science/pith/C56OK6IHPPP2JJ2SJZXMVGYBCX/action/replication_record"}},"created_at":"2026-06-30T02:17:42.988950+00:00","updated_at":"2026-06-30T02:17:42.988950+00:00"}