{
  "id": "math/0611352",
  "version": "v1",
  "published": "2006-11-12T14:41:06.000Z",
  "updated": "2006-11-12T14:41:06.000Z",
  "title": "Exponents of Diophantine Approximation in dimension two",
  "authors": [
    "Michel Laurent"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Theta=(\\alpha,\\beta)$ be a point in $\\bR^2$, with $1,\\alpha,\\beta$ linearly independent over $\\bQ$. We attach to $\\Theta$ a quadruple $\\Omega(\\Theta)$ of exponents which measure the quality of approximation to $\\Theta$ both by rational points and by rational lines. The two ``uniform'' components of $\\Omega(\\Theta)$ are related by an equation, due to Jarn{\\'\\i}k, and the four exponents satisfy two inequalities which refine Khintchine's transference principle. Conversely, we show that for any quadruple $\\Omega$ fulfilling these necessary conditions, there exists a point $\\Theta\\in \\bR^2$ for which $\\Omega(\\Theta) =\\Omega$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-11-12T14:41:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J13",
      "11J70"
    ],
    "keywords": [
      "diophantine approximation",
      "refine khintchines transference principle",
      "exponents satisfy",
      "rational points",
      "rational lines"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11352L"
    }
  }
}