{"paper":{"title":"Diophantine approximation by almost equilateral triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Daniele Mundici","submitted_at":"2017-05-09T14:23:26Z","abstract_excerpt":"A {\\it two-dimensional continued fraction expansion} is a map $\\mu$ assigning to every $x \\in\\mathbb R^2\\setminus\\mathbb Q^2$\n  a sequence $\\mu(x)=T_0,T_1,\\dots$ of triangles $T_n$ with vertices $x_{ni}=(p_{ni}/d_{ni},q_{ni}/d_{ni})\\in\\mathbb Q^2, d_{ni}>0, p_{ni}, q_{ni}, d_{ni}\\in \\mathbb Z,$ $i=1,2,3$, such that \\begin{eqnarray*} \\det \\left(\\begin{matrix} p_{n1}& q_{n1} &d_{n1}\\\\ p_{n2}& q_{n2} &d_{n2}\\\\ p_{n3}& q_{n3} &d_{n3} \\end{matrix} \\right) = \\pm 1\\,\\,\\, \\,\\,\\,\\mbox{and}\\,\\,\\,\\,\\,\\, \\bigcap_n T_n = \\{x\\}. \\end{eqnarray*} We construct a two-dimensional continued fraction expansion $\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.03344","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"}