{
  "id": "1809.09742",
  "version": "v1",
  "published": "2018-09-25T22:09:38.000Z",
  "updated": "2018-09-25T22:09:38.000Z",
  "title": "A Jarník-type theorem for a problem of approximation by cubic polynomials",
  "authors": [
    "Alessandro Pezzoni"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For a given decreasing positive real function $\\psi$, let $\\mathcal{A}_n(\\psi)$ be the set of real numbers for which there are infinitely many integer polynomials $P$ of degree up to $n$ such that $\\left\\lvert P(x) \\right\\rvert \\leq \\psi(\\operatorname{H}(P))$. A theorem by Bernik states that $\\mathcal{A}_n(\\psi)$ has Hausdorff dimension $\\frac{n+1}{w+1}$ in the special case $\\psi(r) = r^{-w}$, while a theorem by Beresnevich, Dickinson and Velani implies that the Hausdorff measure $\\operatorname{\\mathcal{H}}^g(\\mathcal{A}_n(\\psi))=\\infty$ when a certain series diverges. In this paper we prove the convergence counterpart of this result when $P$ has bounded discriminant, which leads to a complete solution when $n = 3$ and $\\psi(r) = r^{-w}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-25T22:09:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J83"
    ],
    "keywords": [
      "cubic polynomials",
      "jarník-type theorem",
      "approximation",
      "complete solution",
      "decreasing positive real function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}