{
  "id": "1707.05094",
  "version": "v1",
  "published": "2017-07-17T11:11:33.000Z",
  "updated": "2017-07-17T11:11:33.000Z",
  "title": "Piatetski-Shapiro sequences via Beatty sequences",
  "authors": [
    "Lukas Spiegelhofer"
  ],
  "comment": "32 pages, published in Acta Arithmetica",
  "journal": "Acta Arith. 166 (2014), no. 3, 201--229",
  "categories": [
    "math.NT"
  ],
  "abstract": "Integer sequences of the form $\\lfloor n^c\\rfloor$, where $1<c<2$, can be locally approximated by sequences of the form $\\lfloor n\\alpha+\\beta\\rfloor$ in a very good way. Following this approach, we are led to an estimate of the difference \\[\\sum_{n\\leq x}\\varphi\\left(\\lfloor n^c\\rfloor\\right)-\\frac 1c\\sum_{n\\leq x^c}\\varphi(n)n^{\\frac 1c-1},\\] which measures the deviation of the mean value of $\\varphi$ on the subsequence $\\lfloor n^c\\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for $1<c\\leq 1.42$ the subsequence of the Thue-Morse sequence indexed by $\\lfloor n^c\\rfloor$ attains both of its values with asymptotic density $1/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-17T11:11:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B83",
      "11A63"
    ],
    "keywords": [
      "beatty sequences",
      "piatetski-shapiro sequences",
      "mean value",
      "integer sequences",
      "subsequence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}