{
  "id": "1405.5762",
  "version": "v2",
  "published": "2014-05-22T14:06:59.000Z",
  "updated": "2014-05-29T11:57:16.000Z",
  "title": "Badly approximable numbers for sequences of balls",
  "authors": [
    "Simon Baker"
  ],
  "comment": "7 pages. After completing this paper the author was made aware of a result due to Cassels. We now know that the work done in this paper is in fact a reasonably straightforward consequence of this result",
  "categories": [
    "math.NT",
    "math.CA",
    "math.DS"
  ],
  "abstract": "It is a classical result from Diophantine approximation that the set of badly approximable numbers has Lebesgue measure zero. In this paper we generalise this result to more general sequences of balls. Given a countable set of closed $d$-dimensional Euclidean balls $\\{B(x_{i},r_{i})\\}_{i=1}^{\\infty},$ we say that $\\alpha\\in \\mathbb{R}^{d}$ is a badly approximable number with respect to $\\{B(x_{i},r_{i})\\}_{i=1}^{\\infty}$ if there exists $\\kappa(\\alpha)>0$ and $N(\\alpha)\\in\\mathbb{N}$ such that $\\alpha\\notin B(x_{i},\\kappa(\\alpha)r_{i})$ for all $i\\geq N(\\alpha)$. Under natural conditions on the set of balls, we prove that the set of badly approximable numbers with respect to $\\{B(x_{i},r_{i})\\}_{i=1}^{\\infty}$ has Lebesgue measure zero. Moreover, our approach yields a new proof that the set of badly approximable numbers has Lebesgue measure zero.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-29T11:57:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K60",
      "28A80"
    ],
    "keywords": [
      "badly approximable number",
      "lebesgue measure zero",
      "dimensional euclidean balls",
      "general sequences",
      "diophantine approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.5762B"
    }
  }
}