{
  "id": "2205.12637",
  "version": "v1",
  "published": "2022-05-25T10:29:01.000Z",
  "updated": "2022-05-25T10:29:01.000Z",
  "title": "A Central Limit Theorem for Counting Functions Related to Symplectic Lattices and Bounded Sets",
  "authors": [
    "Kristian Holm"
  ],
  "categories": [
    "math.NT",
    "math.DS",
    "math.PR"
  ],
  "abstract": "We use a method developed by Bj\\\"orklund and Gorodnik to show a central limit theorem (as $T$ tends to $\\infty$) for the counting functions $\\# \\left( \\Lambda \\cap \\Omega_T \\right)$ where $\\Lambda$ ranges over the space $Y_{2d}$ of symplectic lattices in $\\mathbb{R}^{2d}$ ($d \\geqslant 4$). Here $\\lbrace \\Omega_T \\rbrace_T$ is a certain family of bounded domains in $\\mathbb{R}^{2d}$ that can be tessellated by means of the action of a diagonal semigroup contained in $\\mathrm{Sp}(2d, \\mathbb{R})$. In the process we obtain new $L^p$ bounds on a certain height function on $Y_{2d}$ originally introduced by Schmidt.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-25T10:29:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P21",
      "60F05",
      "37A25"
    ],
    "keywords": [
      "central limit theorem",
      "symplectic lattices",
      "counting functions",
      "bounded sets",
      "diagonal semigroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}