{"paper":{"title":"Erd\\H{o}s--Falconer distance conjecture from an analytic perspective","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dung The Tran, Le Quang Ham","submitted_at":"2026-07-07T07:30:38Z","abstract_excerpt":"Let \\(q\\) be an odd prime power and let \\(V=\\F_q^{2m}\\), equipped with \\(Q(x)=x_1^2+\\cdots+x_{2m}^2\\). We develop a semidefinite Delsarte framework for the two-set Erd\\H{o}s--Falconer distance problem over \\(V\\). The framework reduces the natural \\(q^m\\)-scale positive-proportion theorem to a uniform \\(L^1\\) anti-concentration statement for positive convex combinations of classical Kloosterman sums. Assuming this Kloosterman anti-concentration conjecture, we prove that for every \\(0<\\alpha<\\frac{1}{2}\\) there is a constant \\(C_{m,\\alpha}\\) such that \\[\n  \\min\\{|E|,|F| \\} \\ge C_{m,\\alpha}q^m\n  "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05926","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/2607.05926/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"}