{
  "id": "1405.7650",
  "version": "v3",
  "published": "2014-05-29T18:46:27.000Z",
  "updated": "2015-09-21T11:38:59.000Z",
  "title": "Intrinsic Diophantine approximation on quadric hypersurfaces",
  "authors": [
    "Lior Fishman",
    "Dmitry Kleinbock",
    "Keith Merrill",
    "David Simmons"
  ],
  "comment": "\"Part I: General theory\" from the previous version has been moved to http://arxiv.org/abs/1509.05439. As a result, theorem numbering has changed",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "We consider the question of how well points in a quadric hypersurface $M\\subset\\mathbb R^d$ can be approximated by rational points of $\\mathbb Q^d\\cap M$. This contrasts with the more common setup of approximating points in a manifold by all rational points in $\\mathbb Q^d$. We provide complete answers to major questions of Diophantine approximation in this context. Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated. Our methods include a correspondence between the intrinsic Diophantine approximation theory on a rational quadric hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface, similar to earlier results in the non-intrinsic setting due to Dani ('86) and Kleinbock--Margulis ('99).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-18T06:54:20.000Z",
      "title": "Intrinsic Diophantine Approximation on Manifolds",
      "abstract": "We consider the question of how well points in a manifold $M\\subseteq\\mathbb R^d$ can be approximated by rational points of $\\mathbb Q^d\\cap M$. This contrasts with the more common setup of approximating points in $M$ by all rational points in $\\mathbb Q^d$. Our theorems are of two classes: theorems which apply to all nondegenerate manifolds -- which by their nature must be of the form \"many or most points of $M$ are not very well approximable by rationals\" -- and theorems concerning intrinsic approximation for rational quadratic hypersurfaces. In the latter case, we provide a complete answer to the four major questions of Diophantine approximation in this context. Extending results of Dani ('86) and Kleinbock--Margulis ('99), our methods utilize a correspondence between the intrinsic Diophantine approximation theory on a rational quadratic hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface.",
      "comment": "Updated references and corrected minor typos",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-09-21T11:38:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K60",
      "37A17"
    ],
    "keywords": [
      "rational points",
      "intrinsic diophantine approximation theory",
      "rational quadric hypersurface",
      "common setup",
      "rational ranks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.7650F"
    }
  }
}