{
  "id": "2211.15830",
  "version": "v1",
  "published": "2022-11-28T23:41:17.000Z",
  "updated": "2022-11-28T23:41:17.000Z",
  "title": "On the multiplicative independence between $n$ and $\\lfloor α n\\rfloor$",
  "authors": [
    "David Crnčevíc",
    "Felipe Hernández",
    "Kevin Rizk",
    "Khunpob Sereesuchart",
    "Ran Tao"
  ],
  "comment": "31 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article we investigate different forms of multiplicative independence between the sequences $n$ and $\\lfloor n \\alpha \\rfloor$ for irrational $\\alpha$. Our main theorem shows that for a large class of arithmetic functions $a, b \\colon \\mathbb{N} \\to \\mathbb{C}$ the sequences $(a(n))_{n \\in \\mathbb{N}}$ and $(b ( \\lfloor \\alpha n \\rfloor))_{n \\in \\mathbb{N}}$ are asymptotically uncorrelated. This new theorem is then applied to prove a $2$-dimensional version of the Erd\\H{o}s-Kac theorem, asserting that the sequences $(\\omega(n))_{n \\in \\mathbb{N}}$ and $(\\omega( \\lfloor \\alpha n \\rfloor)_{n\\in \\mathbb{N}}$ behave as independent normally distributed random variables with mean $\\log\\log n$ and standard deviation $\\sqrt{ \\log \\log n}$. Our main result also implies a variation on Chowla's Conjecture asserting that the logarithmic average of $(\\lambda(n) \\lambda ( \\lfloor \\alpha n \\rfloor))_{n \\in \\mathbb{N}}$ tends to $0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-28T23:41:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N60",
      "11N37",
      "11N05"
    ],
    "keywords": [
      "multiplicative independence",
      "independent normally distributed random variables",
      "large class",
      "dimensional version",
      "main theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}