{
  "id": "2312.12043",
  "version": "v1",
  "published": "2023-12-19T10:55:54.000Z",
  "updated": "2023-12-19T10:55:54.000Z",
  "title": "Rational approximations to values of $E$-functions",
  "authors": [
    "Stéphane Fischler",
    "Tanguy Rivoal"
  ],
  "comment": "32 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We solve a long standing problem in the theory of Siegel's $E$-functions, initiated by Lang for Bessel's function $J_0$ in the 60's and considered in full generality by G. Chudnovsky in the 80's: we prove that irrational values taken at rational points by $E$-functions with rational Taylor coefficients have irrationality exponent equal to 2. This result had been obtained before by Zudilin under stronger assumptions on algebraic independence of $E$-functions, satisfied by $J_0$ but not by all hypergeometric $E$-functions for instance.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-19T10:55:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J82",
      "11J91"
    ],
    "keywords": [
      "rational approximations",
      "irrational values taken",
      "rational taylor coefficients",
      "irrationality exponent equal",
      "rational points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}