{
  "id": "1202.1948",
  "version": "v1",
  "published": "2012-02-09T11:02:29.000Z",
  "updated": "2012-02-09T11:02:29.000Z",
  "title": "Purely periodic expansions in systems with negative base",
  "authors": [
    "Zuzana Masáková",
    "Edita Pelantová"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama's result for positive Pisot unit base $\\beta$, we find a sufficient condition so that there exist an interval $J$ containing the origin such that the $(-\\beta)$-expansion of every rational number from $J$ is purely periodic. We focus on the case of quadratic bases and demonstrate the following difference between the negative and positive bases: It is known that the finiteness property (${\\rm Fin}(\\beta)=\\Z[\\beta]$) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that ${\\rm Fin}(-\\beta)=\\Z[\\beta]$ is not necessary in the case of negative bases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-09T11:02:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "purely periodic expansions",
      "negative base numeration system",
      "positive pisot unit base",
      "finiteness property",
      "quadratic bases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.1948M"
    }
  }
}