{
  "id": "1902.09712",
  "version": "v1",
  "published": "2019-02-26T02:46:25.000Z",
  "updated": "2019-02-26T02:46:25.000Z",
  "title": "Sarnak's Conjecture for nilsequences on arbitrary number fields and applications",
  "authors": [
    "Wenbo Sun"
  ],
  "comment": "65 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We formulate the generalized Sarnak's M\\\"obius disjointness conjecture for an arbitrary number field $K$, and prove a quantitative disjointness result between polynomial nilsequences $(\\Phi(g(n)\\Gamma))_{n\\in\\mathbb{Z}^{D}}$ and aperiodic multiplicative functions on $\\mathcal{O}_{K}$, the ring of integers of $K$. Here $D=[K\\colon\\mathbb{Q}]$, $X=G/\\Gamma$ is a nilmanifold, $g\\colon\\mathbb{Z}^{D}\\to G$ is a polynomial sequence, and $\\Phi\\colon X\\to \\mathbb{C}$ is a Lipschitz function. The proof uses tools from multi-dimensional higher order Fourier analysis, multi-linear analysis, orbit properties on nilmanifold, and an orthogonality criterion of K\\'atai in $\\mathcal{O}_{K}$. We also use variations of this result to derive applications in number theory and combinatorics: (1) we prove a structure theorem for multiplicative functions on $K$, saying that every bounded multiplicative function can be decomposed into the sum of an almost periodic function (the structural part) and a function with small Gowers uniformity norm of any degree (the uniform part); (2) we give a necessary and sufficient condition for the Gowers norms of a bounded multiplicative function in $\\mathcal{O}_{K}$ to be zero; (3) we provide partition regularity results over $K$ for a large class of homogeneous equations in three variables. For example, for $a,b\\in\\mathbb{Z}\\backslash\\{0\\}$, we show that for every partition of $\\mathcal{O}_{K}$ into finitely many cells, where $K=\\mathbb{Q}(\\sqrt{a},\\sqrt{b},\\sqrt{a+b})$, there exist distinct and non-zero $x,y$ belonging to the same cell and $z\\in\\mathcal{O}_{K}$ such that $ax^{2}+by^{2}=z^{2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-26T02:46:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "05D10",
      "11B30",
      "11N60",
      "11N80",
      "11R04",
      "37A45"
    ],
    "keywords": [
      "arbitrary number field",
      "sarnaks conjecture",
      "multi-dimensional higher order fourier analysis",
      "nilsequences",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}