{
  "id": "math/0401148",
  "version": "v2",
  "published": "2004-01-14T13:47:11.000Z",
  "updated": "2006-04-26T11:44:10.000Z",
  "title": "Diophantine approximation on planar curves and the distribution of rational points",
  "authors": [
    "Victor Beresnevich",
    "Detta Dickinson",
    "Sanju Velani"
  ],
  "comment": "With an Appendix by Bob Vaughan: Sums of two squares near perfect squares",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\cal C$ be a non--degenerate planar curve and for a real, positive decreasing function $\\psi$ let $\\cal C(\\psi)$ denote the set of simultaneously $\\psi$--approximable points lying on $\\cal C$. We show that $\\cal C$ is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on $\\cal C$ of $\\cal C(\\psi)$ is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that $\\cal C$ is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions $\\psi$ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of $\\cal C(\\psi)$. These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-04-26T11:44:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diophantine approximation",
      "rational points",
      "distribution",
      "divergent khintchine type result",
      "non-degenerate planar curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1148B"
    }
  }
}