{
  "id": "2201.08800",
  "version": "v1",
  "published": "2022-01-21T17:59:10.000Z",
  "updated": "2022-01-21T17:59:10.000Z",
  "title": "Orders of Oscillation Motivated by Sarnak's Conjecture, Part II",
  "authors": [
    "Yunping Jiang"
  ],
  "comment": "19 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "I have investigated orders of oscillating sequences motivated by Sarnak's conjecture in~\\cite{JPAMS} and proved that an oscillating sequence of order $d$ is linearly disjoint from affine distal flows on the $d$-torus. One of the consequences is that an oscillating sequence of order $d$ in the arithmetic sense is linearly disjoint from affine flows with zero topological entropy on the $d$-torus. In this paper, I will extend these results to polynomial skew products on the $d$-torus, that is, given a polynomial skew product on the $d$-torus, there is a positive integer $m$ such that any oscillating sequence of order $m$ is linearly disjoint from this polynomial skew product. In particular, when all polynomials depend only on the first variable, I have that an oscillating sequence of order $m=d+k-1$ is linearly disjoint from all polynomial skew products on the $d$-torus with polynomials of degree less than or equal to $k$. One of the consequences is the linear disjointness for flows which are automorphisms of the $d$-torus with absolute values of eigenvalues $1$ plus a polynomial vector and oscillating sequences of order $m$ in the arithmetic sense. Furthermore, I will prove that an oscillating sequence of order $d$ is linearly disjoint from minimal mean attractable and minimal quasi-discrete spectrum of order $d$ flows. Finally, I define and give some examples of Chowla sequences from our paper~\\cite{AJ}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-21T17:59:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A35",
      "11K65",
      "37A25",
      "11N05"
    ],
    "keywords": [
      "oscillating sequence",
      "polynomial skew product",
      "sarnaks conjecture",
      "linearly disjoint",
      "oscillation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}