{
  "id": "2312.06981",
  "version": "v1",
  "published": "2023-12-12T04:52:24.000Z",
  "updated": "2023-12-12T04:52:24.000Z",
  "title": "Linear independence of series related to the Thue--Morse sequence along powers",
  "authors": [
    "Michael Coons",
    "Yohei Tachiya"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT",
    "cs.FL",
    "math.CO"
  ],
  "abstract": "The Thue--Morse sequence $\\{t(n)\\}_{n\\geqslant 1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive integers $n$, where $t(n)=1$ (resp. $=0$) if the binary expansion of $n$ has an odd (resp. even) number of ones. In this paper, we generalize a recent result of E.~Miyanohara by showing that, for a fixed Pisot or Salem number $\\beta>\\sqrt{\\varphi}=1.272019649\\ldots$, the set of the numbers $$ 1,\\quad \\sum_{n\\geqslant 1}\\frac{t(n)}{\\beta^{n}},\\quad \\sum_{n\\geqslant 1}\\frac{t(n^2)}{\\beta^{n}},\\quad \\dots, \\quad \\sum_{n\\geqslant 1}\\frac{t(n^k)}{\\beta^{n}},\\quad \\dots $$ is linearly independent over the field $\\mathbb{Q}(\\beta)$, where $\\varphi:=(1+\\sqrt{5})/2$ is the golden ratio. Our result implies that for any $k\\geqslant 1$ and for any $a_1,a_2,\\ldots,a_k\\in\\mathbb{Q}(\\beta)$, not all zero, the sequence \\{$a_1t(n)+a_2t(n^2)+\\cdots+a_kt(n^k)\\}_{n\\geqslant 1}$ cannot be eventually periodic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-12T04:52:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J72",
      "11A63",
      "11B85"
    ],
    "keywords": [
      "thue-morse sequence",
      "linear independence",
      "binary expansion",
      "result implies",
      "indicator function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}