{
  "id": "1705.08979",
  "version": "v1",
  "published": "2017-05-24T22:09:11.000Z",
  "updated": "2017-05-24T22:09:11.000Z",
  "title": "Automatic sequences and generalised polynomials",
  "authors": [
    "Jakub Byszewski",
    "Jakub Konieczny"
  ],
  "comment": "28 pages, an extended version of the second half of arxiv:1610.03900 [math.NT]",
  "categories": [
    "math.NT",
    "cs.FL",
    "math.CO",
    "math.DS"
  ],
  "abstract": "We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are periodic away from a finite set. Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial $p(n)$ with at least one irrational coefficient (except for the constant one) and integer $m$, the sequence $\\lfloor p(n)\\rfloor \\bmod{m}$ is never automatic. We also obtain a conditional result, where we prove the conjecture under the assumption that the characteristic sequence of the set of powers of an integer $k\\geq 2$ is not given by a generalised polynomial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-24T22:09:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B85",
      "37A45",
      "37B05",
      "37B10",
      "11J71",
      "11B37",
      "05C20"
    ],
    "keywords": [
      "automatic sequences",
      "periodic away",
      "conjecture",
      "finite automata",
      "trivial case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}