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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"},"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":"2608.01274","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-08-02T14:32:48Z","cross_cats_sorted":["math.CA","math.CO"],"title_canon_sha256":"50b7fffe696cd42cc2ba0d52e48eb9f437bad42915e270c355fa20800b7d8121","abstract_canon_sha256":"67d3bd69d591d4ba0933ca33977625ac70552cc12fcf6f69c1ef663dfc73afbe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:02:47.617746Z","signature_b64":"q2gy+Z15dZDqGXT1BM8CZaeQlPlcyH87IZ8PwQw2YQva0DnV1Y3YVdgRCIljVYX9/15VdlzzK/7fq5p4oTa2Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c71add78ed37fabc89c2e800ca86af1732bebc21dbfff1e7fbe99efac91ac68f","last_reissued_at":"2026-08-04T02:02:47.616286Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:02:47.616286Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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