{
  "id": "2403.04159",
  "version": "v1",
  "published": "2024-03-07T02:31:57.000Z",
  "updated": "2024-03-07T02:31:57.000Z",
  "title": "Borel-Bernstein theorem and Hausdorff dimension of sets in power-2-decaying Gauss-like expansion",
  "authors": [
    "Zhihui Li",
    "Xin Liao",
    "Dingding Yu"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Each $x\\in (0,1]$ can be uniquely expanded as a power-2-decaying Gauss-like expansion, in the form of $$ x=\\sum_{i=1}^{\\infty}2^{-(d_1(x)+d_2(x)+\\cdots+d_i(x))},\\qquad d_i(x)\\in \\mathbb{N}. $$ Let $\\phi:\\mathbb{N}\\to \\mathbb{R}^{+}$ be an arbitrary positive function. We are interested in the size of the set $$F(\\phi)=\\{x\\in (0,1]:d_n(x)\\ge \\phi(n)~~\\text{i.m.}~n\\}.$$ We prove a Borel-Bernstein theorem on the zero-one law of the Lebesgue measure of $F(\\phi)$. We also obtain the Hausdorff dimension of $F(\\phi)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-07T02:31:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hausdorff dimension",
      "borel-bernstein theorem",
      "gauss-like expansion",
      "arbitrary positive function",
      "zero-one law"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}