{"paper":{"title":"On the quotient set of the distance set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.CA","authors_text":"A. Iosevich, D. Koh, H. Parshall","submitted_at":"2018-02-22T20:49:40Z","abstract_excerpt":"Let ${\\Bbb F}_q$ be a finite field of order $q.$ We prove that if $d\\ge 2$ is even and $E \\subset {\\Bbb F}_q^d$ with $|E| \\ge 9q^{\\frac{d}{2}}$ then $$ {\\Bbb F}_q=\\frac{\\Delta(E)}{\\Delta(E)}=\\left\\{ \\frac{a}{b}: a \\in \\Delta(E), b \\in \\Delta(E) \\backslash \\{0\\} \\right\\},$$ where $$ \\Delta(E)=\\{||x-y||: x,y \\in E\\}, \\ ||x||=x_1^2+x_2^2+\\cdots+x_d^2.$$ If the dimension $d$ is odd and $E\\subset \\mathbb F_q^d$ with $|E|\\ge 6q^{\\frac{d}{2}},$ then $$ \\{0\\}\\cup\\mathbb F_q^+ \\subset \\frac{\\Delta(E)}{\\Delta(E)},$$ where $\\mathbb F_q^+$ denotes the set of nonzero quadratic residues in $\\mathbb F_q.$ Bo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.08297","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":""},"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"}