{
  "id": "1511.01671",
  "version": "v1",
  "published": "2015-11-05T09:38:19.000Z",
  "updated": "2015-11-05T09:38:19.000Z",
  "title": "Normality of the Thue--Morse sequence along Piatetski-Shapiro sequences, II",
  "authors": [
    "Clemens Müllner",
    "Lukas Spiegelhofer"
  ],
  "comment": "35 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that the Thue--Morse sequence $\\mathbf t$ along subsequences indexed by $\\lfloor n^c\\rfloor$ is normal, where $1<c<3/2$. That is, for $c$ in this range and for each $\\omega\\in\\{0,1\\}^L$, where $L\\geq 1$, the set of occurrences of $\\omega$ as a subword (contiguous finite subsequence) of the sequence $n\\mapsto \\mathbf t_{\\lfloor n^c\\rfloor}$ has asymptotic density $2^{-L}$. This is an improvement over a recent result by the second author, which handles the case $1<c<4/3$. In particular, this result shows that for $1<c<3/2$ the sequence $n\\mapsto \\mathbf t_{\\lfloor n^c\\rfloor}$ attains both of its values with asymptotic density $1/2$, which improves on the bound $c<1.4$ obtained by Mauduit and Rivat (who obtained this bound in the more general setting of $q$-multiplicative functions, however) and on the bound $c\\leq 1.42$ obtained by the second author. In the course of proving the main theorem, we show that $2/3$ is an admissible level of distribution for the Thue--Morse sequence, that is, it satisfies a Bombieri--Vinogradov type theorem for each exponent $\\eta<2/3$. This improves on a result by Fouvry and Mauduit, who obtained the exponent $0.5924$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-05T09:38:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K16",
      "11A63",
      "11B25",
      "11B83"
    ],
    "keywords": [
      "thue-morse sequence",
      "piatetski-shapiro sequences",
      "second author",
      "asymptotic density",
      "bombieri-vinogradov type theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151101671M"
    }
  }
}