{
  "id": "2405.08341",
  "version": "v1",
  "published": "2024-05-14T06:27:44.000Z",
  "updated": "2024-05-14T06:27:44.000Z",
  "title": "On approximation to a real number by algebraic numbers of bounded degree",
  "authors": [
    "Anthony Poëls"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In his seminal 1961 paper, Wirsing studied how well a given transcendental real number $\\xi$ can be approximated by algebraic numbers $\\alpha$ of degree at most $n$ for a given positive integer $n$, in terms of the so-called naive height $H(\\alpha)$ of $\\alpha$. He showed that the infimum $\\omega^*_n(\\xi)$ of all $\\omega$ for which infinitely many such $\\alpha$ have $|\\xi-\\alpha| \\le H(\\alpha)^{-\\omega-1}$ is at least $(n+1)/2$. He also asked if we could even have $\\omega^*_n(\\xi) \\ge n$ as it is generally expected. Since then, all improvements on Wirsing's lower bound were of the form $n/2+\\mathcal{O}(1)$ until Badziahin and Schleischitz showed in 2021 that $\\omega^*_n(\\xi) \\ge an$ for each $n\\ge 4$, with $a=1/\\sqrt{3}\\simeq 0.577$. In this paper, we use a different approach partly inspired by parametric geometry of numbers and show that $\\omega^*_n(\\xi) \\ge an$ for each $n\\ge 2$, with $a=1/(2-\\log 2)\\simeq 0.765$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-14T06:27:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J13",
      "11J82"
    ],
    "keywords": [
      "algebraic numbers",
      "bounded degree",
      "approximation",
      "transcendental real number",
      "wirsings lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}