{
  "id": "1311.1106",
  "version": "v4",
  "published": "2013-11-05T16:09:47.000Z",
  "updated": "2015-11-08T12:58:21.000Z",
  "title": "Equidistribution of expanding curves in homogeneous spaces and Diophantine approximation for square matrices",
  "authors": [
    "Lei Yang"
  ],
  "comment": "14 pages. The paper is rewritten according to referee's suggestions. An appendix is added. arXiv admin note: text overlap with arXiv:1303.6023",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "In this article, we study an analytic curve $\\varphi: I=[a,b]\\rightarrow \\mathrm{M}(n\\times n, \\mathbb{R})$ in the space of $n$ by $n$ real matrices, and show that if $\\varphi$ satisfies certain geometric conditions, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem is not improvable. To do this, we embed the curve into some homogeneous space $G/\\Gamma$, and prove that under the action of some expanding diagonal flow $A= \\{a(t): t \\in \\mathbb{R}\\}$, the expanding curves tend to be equidistributed in $G/\\Gamma$, as $t \\rightarrow +\\infty$. This solves a special case of a problem proposed by Nimish Shah in ~\\cite{Shah_1}.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-11-29T21:57:42.000Z",
      "title": "Limit distributions of curves in homogeneous spaces and Diophantine approximation",
      "abstract": "In this paper, we consider an analytical curve $\\varphi: I=[a,b] \\rightarrow \\mathrm{M}(n\\times n,\\mathbb{R})$ with invertible derivative, and will prove that unless $\\{(\\varphi(s)-\\varphi(s_0))^{-1} : s\\in I\\}$ is contained in some proper affine subspace of $\\mathrm{M}(n\\times n,\\mathbb{R})$ for all $s_0 \\in I$ and $s$ in some neighborhood of $s_0$, for almost all points on the curve, the Dirichlet's theorem can not be improved. We prove this result by embedding the curve into the homogeneous space $\\mathrm{SL}(2n,\\mathbb{R})/\\mathrm{SL}(2n,\\mathbb{Z})$ and investigate the limit distribution of the curve under some expanding diagonal element. The proof heavily depends on properties of representation of $\\mathrm{SL}(2,\\mathbb{R})$.",
      "comment": "14 page, some minor corrections are made. arXiv admin note: text overlap with arXiv:1303.6023",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-11-08T12:58:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A17",
      "22F30",
      "11J13"
    ],
    "keywords": [
      "limit distribution",
      "homogeneous space",
      "diophantine approximation",
      "proper affine subspace",
      "dirichlets theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.1106Y"
    }
  }
}