{
  "id": "2311.06289",
  "version": "v1",
  "published": "2023-11-03T04:25:24.000Z",
  "updated": "2023-11-03T04:25:24.000Z",
  "title": "On some properties of Perron numbers",
  "authors": [
    "Nikita Sidorov"
  ],
  "comment": "two pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\theta$ be a real number, $n\\in\\mathbb N$, and $D_n(\\theta)=\\left\\{\\sum_{k=1}^n a_k\\theta^{k}\\mid a_k\\in\\{0,\\dots,\\lfloor \\theta\\rfloor\\}\\right\\}. $ Let $\\theta$ be a Perron number, that is, an algebraic integer $>1$ whose other Galois conjugates are less than $\\theta$ in absolute value. I shall prove two results: (1) $\\theta^n\\ll\\#D_n(\\theta)\\ll\\sqrt n \\theta^n.$ (2) $\\theta$ is of height $\\le \\lfloor\\theta\\rfloor$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-03T04:25:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R06"
    ],
    "keywords": [
      "perron number",
      "properties",
      "real number",
      "algebraic integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}