{"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"}