{
  "id": "1705.03344",
  "version": "v1",
  "published": "2017-05-09T14:23:26.000Z",
  "updated": "2017-05-09T14:23:26.000Z",
  "title": "Diophantine approximation by almost equilateral triangles",
  "authors": [
    "Daniele Mundici"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "A {\\it two-dimensional continued fraction expansion} is a map $\\mu$ assigning to every $x \\in\\mathbb R^2\\setminus\\mathbb Q^2$ 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 $\\mu^*$ such that for densely many (Turing computable) points $x$ the vertices of the triangles of $\\mu(x)$ strongly converge to $x$. Strong convergence depends on the value of $\\lim_{n\\to \\infty}\\frac{\\sum_{i=1}^3\\dist(x,x_{ni})}{(2d_{n1}d_{n2}d_{n3})^{-1/2}},$ (\"dist\" denoting euclidean distance) which in turn depends on the smallest angle of $T_n$. Our proofs combine a classical theorem of Davenport Mahler in diophantine approximation, with the algorithmic resolution of toric singularities in the equivalent framework of regular fans and their stellar operations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-09T14:23:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A55"
    ],
    "keywords": [
      "diophantine approximation",
      "equilateral triangles",
      "two-dimensional continued fraction expansion",
      "stellar operations",
      "strong convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}