{
  "id": "1910.08303",
  "version": "v1",
  "published": "2019-10-18T08:10:42.000Z",
  "updated": "2019-10-18T08:10:42.000Z",
  "title": "Multiplication on uniform $λ$-Cantor sets",
  "authors": [
    "Jiangwen Gu",
    "Kan Jiang",
    "Lifeng Xi",
    "Bing Zhao"
  ],
  "comment": "9 pages",
  "categories": [
    "math.DS",
    "math.MG",
    "math.NT"
  ],
  "abstract": "Let $C$ be the middle-third Cantor set. Define $C*C=\\{x*y:x,y\\in C\\}$, where $*=+,-,\\cdot,\\div$ (when $*=\\div$, we assume $y\\neq0$). Steinhaus \\cite{HS} proved in 1917 that \\[ C-C=[-1,1], C+C=[0,2]. \\] In 2019, Athreya, Reznick and Tyson \\cite{Tyson} proved that \\[ C\\div C=\\bigcup_{n=-\\infty}^{\\infty}\\left[ 3^{-n}\\dfrac{2}{3},3^{-n}\\dfrac {3}{2}\\right] . \\] In this paper, we give a description of the topological structure and Lebesgue measure of $C\\cdot C$. We indeed obtain corresponding results on the uniform $\\lambda$-Cantor sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-18T08:10:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiplication",
      "middle-third cantor set",
      "lebesgue measure",
      "description",
      "topological structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}