{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:QEVT4FGSAV64IEARDAXNC4P2I7","short_pith_number":"pith:QEVT4FGS","schema_version":"1.0","canonical_sha256":"812b3e14d2057dc41011182ed171fa47ec1c9f30e81dc4b6ba151ec8be4f76a4","source":{"kind":"arxiv","id":"1705.03344","version":1},"attestation_state":"computed","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 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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 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