{
  "id": "2206.01810",
  "version": "v1",
  "published": "2022-06-03T20:34:50.000Z",
  "updated": "2022-06-03T20:34:50.000Z",
  "title": "On Periodic Alternate Base Expansions",
  "authors": [
    "Émilie Charlier",
    "Célia Cisternino",
    "Savinien Kreczman"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For an alternate base $\\boldsymbol{\\beta}=(\\beta_0,\\ldots,\\beta_{p-1})$, we show that if all rational numbers in the unit interval $[0,1)$ have periodic alternate expansions with respect to the $p$ shifts of $\\boldsymbol{\\beta}$, then the bases $\\beta_0,\\ldots,\\beta_{p-1}$ all belong to the extension field $\\mathbb{Q}(\\beta)$ where $\\beta$ is the product $\\beta_0\\cdots\\beta_{p-1}$ and moreover, this product $\\beta$ must be either a Pisot number or a Salem number. We also prove the stronger statement that if the bases $\\beta_0,\\ldots,\\beta_{p-1}$ belong to $\\mathbb{Q}(\\beta)$ but the product $\\beta$ is neither a Pisot number nor a Salem number then the set of rationals having an ultimately periodic $\\boldsymbol{\\beta}$-expansion is nowhere dense in $[0,1)$. Moreover, in the case where the product $\\beta$ is a Pisot number and the bases $\\beta_0,\\ldots,\\beta_{p-1}$ all belong to $\\mathbb{Q}(\\beta)$, we prove that the set of points in $[0,1)$ having an ultimately periodic $\\boldsymbol{\\beta}$-expansion is precisely the set $\\mathbb{Q}(\\beta)\\cap[0,1)$. For the restricted case of R\\'enyi real bases, i.e., for $p=1$ in our setting, our method gives rise to an elementary proof of Schmidt's original result. We also give two applications of these results. First, if $\\boldsymbol{\\beta}=(\\beta_0,\\ldots,\\beta_{p-1})$ is an alternate base such that the product $\\beta$ of the bases is a Pisot number and $\\beta_0,\\ldots,\\beta_{p-1}\\in\\mathbb{Q}(\\beta)$, then $\\boldsymbol{\\beta}$ is a Parry alternate base, meaning that the quasi-greedy expansions of $1$ with respect to the $p$ shifts of the base $\\boldsymbol{\\beta}$ are ultimately periodic. Second, we obtain a property of Pisot numbers, namely that if $\\beta$ is a Pisot number then $\\beta\\in\\mathbb{Q}(\\beta^p)$ for all integers $p\\ge 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-03T20:34:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K16"
    ],
    "keywords": [
      "periodic alternate base expansions",
      "pisot number",
      "ultimately periodic",
      "salem number",
      "parry alternate base"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}