{
  "id": "2301.08825",
  "version": "v1",
  "published": "2023-01-20T23:16:16.000Z",
  "updated": "2023-01-20T23:16:16.000Z",
  "title": "Bounding the Largest Inhomogeneous Approximation Constant",
  "authors": [
    "Bishnu Paudel",
    "Chris Pinner"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For a given irrational number $\\alpha$ and a real number $\\gamma$ in $(0,1)$ one defines the two-sided inhomogeneous approximation constant \\begin{equation*} M(\\alpha,\\gamma):=\\liminf_{|n|\\rightarrow\\infty}|n| ||n\\alpha-\\gamma||, \\end{equation*} and the case of worst inhomogeneous approximation for $\\alpha$ \\begin{equation*} \\rho(\\alpha):=\\sup_{\\gamma\\notin\\mathbb{Z}+\\alpha\\mathbb{Z}}M(\\alpha,\\gamma). \\end{equation*} We are interested in lower bounds on $\\rho(\\alpha)$ in terms of $R:=\\liminf_{i\\rightarrow\\infty}a_i,$ where the $a_i$ are the partial quotients in the negative (i.e.\\ the `round-up') continued fraction expansion of $\\alpha$. We obtain bounds for any $R\\geq 3$ which are best possible when $R$ is even (and asymptotically precise when $R$ is odd). In particular when $R\\geq 3$ $$ \\rho(\\alpha)\\geq \\cfrac{1}{6\\sqrt{3}+8}=\\cfrac{1}{18.3923\\dots}, $$ and when $R\\geq 4$, optimally, $$ \\rho(\\alpha) \\geq \\cfrac{1}{4\\sqrt{3}+2}=\\cfrac{1}{8.9282\\ldots}. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-20T23:16:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J20",
      "11J06",
      "11J70"
    ],
    "keywords": [
      "largest inhomogeneous approximation constant",
      "irrational number",
      "real number",
      "two-sided inhomogeneous approximation constant",
      "worst inhomogeneous approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}