{
  "id": "2505.03170",
  "version": "v1",
  "published": "2025-05-06T04:31:33.000Z",
  "updated": "2025-05-06T04:31:33.000Z",
  "title": "The algebraic difference of a Cantor set and its complement",
  "authors": [
    "Piotr Nowakowski",
    "Cheng-Han Pan"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\mathcal{C}\\subseteq[0,1]$ be a Cantor set. In the classical $\\mathcal{C}\\pm\\mathcal{C}$ problems, modifying the ``size'' of $\\mathcal{C}$ has a magnified effect on $\\mathcal{C}\\pm\\mathcal{C}$. However, any gain in $\\mathcal{C}$ necessarily results in a loss in $\\mathcal{C}^c$, and vice versa. This interplay between $\\mathcal{C}$ and its complement $\\mathcal{C}^c$ raises interesting questions about the delicate balance between the two, particularly in how it influences the ``size'' of $\\mathcal{C}^c-\\mathcal{C}$. One of our main results indicates that the Lebesgue measure of $\\mathcal{C}^c-\\mathcal{C}$ has a greatest lower bound of $\\frac{3}{2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-06T04:31:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A05",
      "28A80"
    ],
    "keywords": [
      "cantor set",
      "algebraic difference",
      "complement",
      "greatest lower bound",
      "vice versa"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}