{
  "id": "2411.05770",
  "version": "v1",
  "published": "2024-11-08T18:32:22.000Z",
  "updated": "2024-11-08T18:32:22.000Z",
  "title": "Higher uniformity of arithmetic functions in short intervals II. Almost all intervals",
  "authors": [
    "Kaisa Matomäki",
    "Maksym Radziwiłł",
    "Xuancheng Shao",
    "Terence Tao",
    "Joni Teräväinen"
  ],
  "comment": "104 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study higher uniformity properties of the von Mangoldt function $\\Lambda$, the M\\\"obius function $\\mu$, and the divisor functions $d_k$ on short intervals $(x,x+H]$ for almost all $x \\in [X, 2X]$. Let $\\Lambda^\\sharp$ and $d_k^\\sharp$ be suitable approximants of $\\Lambda$ and $d_k$, $G/\\Gamma$ a filtered nilmanifold, and $F\\colon G/\\Gamma \\to \\mathbb{C}$ a Lipschitz function. Then our results imply for instance that when $X^{1/3+\\varepsilon} \\leq H \\leq X$ we have, for almost all $x \\in [X, 2X]$, \\[ \\sup_{g \\in \\text{Poly}(\\mathbb{Z} \\to G)} \\left| \\sum_{x < n \\leq x+H} (\\Lambda(n)-\\Lambda^\\sharp(n)) \\overline{F}(g(n)\\Gamma) \\right| \\ll H\\log^{-A} X \\] for any fixed $A>0$, and that when $X^{\\varepsilon} \\leq H \\leq X$ we have, for almost all $x \\in [X, 2X]$, \\[ \\sup_{g \\in \\text{Poly}(\\mathbb{Z} \\to G)} \\left| \\sum_{x < n \\leq x+H} (d_k(n)-d_k^\\sharp(n)) \\overline{F}(g(n)\\Gamma) \\right| = o(H \\log^{k-1} X). \\] As a consequence, we show that the short interval Gowers norms $\\|\\Lambda-\\Lambda^\\sharp\\|_{U^s(X,X+H]}$ and $\\|d_k-d_k^\\sharp\\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ in the same ranges of $H$. This in turn allows us to establish the Hardy-Littlewood conjecture and the divisor correlation conjecture with a short average over one variable. Our main new ingredients are type $II$ estimates obtained by developing a \"contagion lemma\" for nilsequences and then using this to \"scale up\" an approximate functional equation for the nilsequence to a larger scale. This extends an approach developed by Walsh for Fourier uniformity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-08T18:32:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11B30"
    ],
    "keywords": [
      "arithmetic functions",
      "study higher uniformity properties",
      "short interval gowers norms",
      "von mangoldt function",
      "divisor correlation conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 104,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}