{
  "id": "2402.03276",
  "version": "v2",
  "published": "2024-02-05T18:34:02.000Z",
  "updated": "2024-02-08T16:15:31.000Z",
  "title": "An approximation of the Collatz map and a lower bound for the average total stopping time",
  "authors": [
    "Manuel Inselmann"
  ],
  "comment": "Some minor corrections and adjustments, in Corollary 2.25 the assumption is replaced with a slightly weaker one",
  "categories": [
    "math.DS",
    "math.CO",
    "math.NT",
    "math.PR"
  ],
  "abstract": "Define the (accelerated) Collatz map on the positive integers by $\\mathsf{Col}_2(n)=\\frac{n}{2}$ if $n$ is even and $\\mathsf{Col}_2(n)=\\frac{3n+1}{2}$ if $n$ is odd. We show that $\\mathsf{Col}_2$ can be approximated by multiplication with $\\frac{3^{\\frac{1}{2}}}{2}$ in the sense that the set of $n$ for which $(\\frac{3^{\\frac{1}{2}}}{2})^kn^{1-\\epsilon}\\leq \\mathsf{Col}_2^k(n)\\leq (\\frac{3^{\\frac{1}{2}}}{2})^kn^{1+\\epsilon}$ for all $0\\leq k\\leq 2(\\log\\frac{4}{3})^{-1}\\log n\\approx 6.952\\log n$ has natural density $1$ for every $\\epsilon>0$. Let $\\tau(n)$ be the minimal $k\\in\\mathbb{N}$ for which $\\mathsf{Col}_2^k(n)=1$ if there exist such a $k$ and set $\\tau(n)=\\infty$ otherwise. As an application of the above we show that $\\liminf_{x\\rightarrow\\infty}\\frac{1}{x\\log x}\\sum_{m=1}^{\\lfloor x\\rfloor}\\tau(m)\\geq 2(\\log\\frac{4}{3})^{-1}$, partially answering a question of Crandall and Shanks. We show also that assuming the Collatz Conjecture is true in the strong sense that $\\tau(n)\\in O(\\log n)$, then $\\lim_{x\\rightarrow\\infty}\\frac{1}{x\\log x}\\sum_{m=1}^{\\lfloor x\\rfloor}\\tau(m)= 2(\\log\\frac{4}{3})^{-1}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-02-08T16:15:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "average total stopping time",
      "collatz map",
      "lower bound",
      "approximation",
      "collatz conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}