{
  "id": "2010.01511",
  "version": "v1",
  "published": "2020-10-04T08:29:41.000Z",
  "updated": "2020-10-04T08:29:41.000Z",
  "title": "Conjugates of Pisot numbers",
  "authors": [
    "Kevin G. Hare",
    "Nikita Sidorov"
  ],
  "comment": "17 pages, 2 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we investigate the Galois conjugates of a Pisot number $q \\in (m, m+1)$, $m \\geq 1$. In particular, we conjecture that for $q \\in (1,2)$ we have $|q'| \\geq \\frac{\\sqrt{5}-1}{2}$ for all conjugates $q'$ of $q$. Further, for $m \\geq 3$, we conjecture that for all Pisot numbers $q \\in (m, m+1)$ we have $|q'| \\geq \\frac{m+1-\\sqrt{m^2+2m-3}}{2}$. A similar conjecture if made for $m =2$. We conjecture that all of these bounds are tight. We provide partial supporting evidence for this conjecture. This evidence is both of a theoretical and computational nature. Lastly, we connect this conjecture to a result on the dimension of Bernoulli convolutions parameterized by $\\beta$, whose conjugate is the reciprocal of a Pisot number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-04T08:29:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K16"
    ],
    "keywords": [
      "pisot number",
      "galois conjugates",
      "partial supporting evidence",
      "computational nature",
      "bernoulli convolutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}