{
  "id": "1308.0954",
  "version": "v1",
  "published": "2013-08-05T12:24:35.000Z",
  "updated": "2013-08-05T12:24:35.000Z",
  "title": "Lattice point counting and height bounds over number fields and quaternion algebras",
  "authors": [
    "Lenny Fukshansky",
    "Glenn Henshaw"
  ],
  "comment": "22 pages; to appear in the Online Journal of Analytic Combinatorics",
  "journal": "Online Journal of Analytic Combinatorics, vol. 8 (2013), art. 4, 20 pp",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a collection of height comparison inequalities for heights over a number field and over a quaternion algebra. We also show how these inequalities can be used to obtain existence results for points of bounded height over a quaternion algebra, which constitute non-commutative analogues of variations of the classical Siegel's lemma and Cassels' theorem on small zeros of quadratic forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-05T12:24:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11H06",
      "52C07",
      "11G50",
      "11E12",
      "11E39"
    ],
    "keywords": [
      "number field",
      "lattice point counting",
      "height bounds",
      "bounded height",
      "numerous applications throughout mathematics"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.0954F"
    }
  }
}