{
  "id": "0909.3324",
  "version": "v3",
  "published": "2009-09-17T21:42:37.000Z",
  "updated": "2011-07-18T15:45:31.000Z",
  "title": "On the topology of sums in powers of an algebraic number",
  "authors": [
    "Nikita Sidorov",
    "Boris Solomyak"
  ],
  "comment": "10 pages, no figures",
  "journal": "Acta Arithmetica 149 (2011), 337-346",
  "doi": "10.4064/aa149-4-3",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $1<q<2$ and \\[ \\Lambda(q)={\\sum_{k=0}^n a_kq^k\\mid a_k\\in\\{-1,0,1\\}, n\\ge1}. \\] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\\pm1$, then $\\Lambda(q)$ is dense in $\\mathbb{R}$. We give several sufficient conditions for the denseness of $\\Lambda(q)$ when $q$ is a root of such a polynomial. In particular, we prove that if $q$ is not a Perron number or it has a conjugate $\\alpha$ such that $q|\\alpha|<1$, then $\\Lambda(q)$ is dense in $\\mathbb{R}$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-07-18T15:45:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J17",
      "11K16",
      "11R06"
    ],
    "keywords": [
      "algebraic number",
      "polynomial",
      "perron number",
      "sufficient conditions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.3324S"
    }
  }
}