{"paper":{"title":"Distribution of simplices in the discrete and continuous settings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.NT","authors_text":"Boqing Xue, Chun-Yen Shen, Thang Pham","submitted_at":"2026-08-02T14:32:48Z","abstract_excerpt":"In this paper, we study the distribution of simplices in both discrete and continuous settings. Let $q$ be an odd prime power, let $Q$ be a nondegenerate quadratic form on $\\mathbb F_q^d$, and let $2\\leq k\\leq d-1$. We prove that every set $E\\subset\\mathbb F_q^d$ with \\[\n  |E|\\geq C_{d,k}q^{\\beta_{d,k}},\n  \\qquad\n  \\beta_{d,k}=\n  \\begin{cases}\n  \\displaystyle \\frac{d+k}{2}-\\frac{k-1}{k+1},\n  & d-k\\ \\text{even},\\\\[2mm]\n  \\displaystyle \\frac{d+k-1}{2},\n  & d-k\\ \\text{odd},\n  \\end{cases} \\] determines a positive proportion of all ordered nondegenerate $k$-simplex congruence classes. This improves"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01274","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/2608.01274/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"}