{
  "id": "2102.07722",
  "version": "v1",
  "published": "2021-02-15T18:16:57.000Z",
  "updated": "2021-02-15T18:16:57.000Z",
  "title": "Expansions in Cantor real bases",
  "authors": [
    "Émilie Charlier",
    "Célia Cisternino"
  ],
  "comment": "21 pages, 3 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "We introduce and study series expansions of real numbers with an arbitrary Cantor real base $\\boldsymbol{\\beta}=(\\beta_n)_{n\\in\\mathbb{N}}$, which we call $\\boldsymbol{\\beta}$-representations. In doing so, we generalize both representations of real numbers in real bases and through Cantor series. We show fundamental properties of $\\boldsymbol{\\beta}$-representations, each of which extends existing results on representations in a real base. In particular, we prove a generalization of Parry's theorem characterizing sequences of nonnegative integers that are the greedy $\\boldsymbol{\\beta}$-representations of some real number in the interval $[0,1)$. We pay special attention to periodic Cantor real bases, which we call alternate bases. In this case, we show that the $\\boldsymbol{\\beta}$-shift is sofic if and only if all quasi-greedy $\\boldsymbol{\\beta}^{(i)}$-expansions of $1$ are ultimately periodic, where $\\boldsymbol{\\beta}^{(i)}$ is the $i$-th shift of the Cantor real base $\\boldsymbol{\\beta}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-15T18:16:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63"
    ],
    "keywords": [
      "real number",
      "representations",
      "arbitrary cantor real base",
      "periodic cantor real bases",
      "study series expansions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}