{
  "id": "2006.13930",
  "version": "v1",
  "published": "2020-06-24T17:58:48.000Z",
  "updated": "2020-06-24T17:58:48.000Z",
  "title": "Distributions of Arithmetic Progressions in Piatetski-Shapiro Sequence",
  "authors": [
    "Kota Saito",
    "Yuuya Yoshida"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "It is known that for all $\\alpha\\in(1,2)$ and all integers $k\\ge3$ and $r\\ge1$, there exist infinitely many $n\\in\\mathbb{N}$ such that the sequence $(\\lfloor{(n+rj)^\\alpha}\\rfloor)_{j=0}^{k-1}$ is an arithmetic progression of length $k$. In this paper, we show that the asymptotic density of all the above $n$ is equal to $1/(k-1)$. Although the common difference $r$ is arbitrarily fixed in the above result, we also examine the case when $r$ is not fixed. Furthermore, we also examine the number of the above $n$ that are contained in a short interval.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-24T17:58:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B25",
      "11B30"
    ],
    "keywords": [
      "arithmetic progression",
      "piatetski-shapiro sequence",
      "distributions",
      "short interval",
      "common difference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}