{
  "id": "1609.08207",
  "version": "v1",
  "published": "2016-09-26T22:10:24.000Z",
  "updated": "2016-09-26T22:10:24.000Z",
  "title": "On the rate of convergence for $(\\log_b n)$",
  "authors": [
    "Chuang Xu"
  ],
  "comment": "20pages",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "In this paper, we study rate of convergence for the distribution of sequence of logarithms $(\\log_bn)$ for integer base $b\\ge2.$ It is well-known that the slowly growing sequence $(\\log_bn)$ is not uniformly distributed modulo one. Its distributions converge to a loop of translated exponential distributions with constant $\\log b$ in the space of probabilities on the circle. We give an upper bound for the rate of convergence under Kantorovich distance $d_{\\mathbb{T}}$ on the circle. We also give a sharp rate of convergence under Kantorovich distance $d_{\\mathbb{R}}$ on the line $\\mathbb{R}.$ It turns out that the convergence under $d_{\\mathbb{T}}$ is much faster than that under $d_{\\mathbb{R}}.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-26T22:10:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K06",
      "11K31"
    ],
    "keywords": [
      "convergence",
      "kantorovich distance",
      "integer base",
      "distributions converge",
      "sharp rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}