{
  "id": "1710.05737",
  "version": "v1",
  "published": "2017-10-16T14:33:23.000Z",
  "updated": "2017-10-16T14:33:23.000Z",
  "title": "Cellular Automata and Powers of $p/q$",
  "authors": [
    "Jarkko Kari",
    "Johan Kopra"
  ],
  "comment": "15 pages, 8 figures. Accepted for publication in RAIRO-ITA",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "We consider one-dimensional cellular automata $F_{p,q}$ which multiply numbers by $p/q$ in base $pq$ for relatively prime integers $p$ and $q$. By studying the structure of traces with respect to $F_{p,q}$ we show that for $p\\geq 2q-1$ (and then as a simple corollary for $p>q>1$) there are arbitrarily small finite unions of intervals which contain the fractional parts of the sequence $\\xi(p/q)^n$, ($n=0,1,2,\\dots$) for some $\\xi>0$. To the other direction, by studying the measure theoretical properties of $F_{p,q}$, we show that for $p>q>1$ there are finite unions of intervals approximating the unit interval arbitrarily well which don't contain the fractional parts of the whole sequence $\\xi(p/q)^n$ for any $\\xi>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-16T14:33:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J71",
      "37A25",
      "68Q80"
    ],
    "keywords": [
      "fractional parts",
      "one-dimensional cellular automata",
      "arbitrarily small finite unions",
      "dont contain",
      "simple corollary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}