{
  "id": "2204.03754",
  "version": "v1",
  "published": "2022-04-07T21:58:46.000Z",
  "updated": "2022-04-07T21:58:46.000Z",
  "title": "Higher uniformity of arithmetic functions in short intervals I. All intervals",
  "authors": [
    "Kaisa Matomäki",
    "Xuancheng Shao",
    "Terence Tao",
    "Joni Teräväinen"
  ],
  "comment": "86 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study higher uniformity properties of the M\\\"obius function $\\mu$, the von Mangoldt function $\\Lambda$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\\theta+\\varepsilon} \\leq H \\leq X^{1-\\varepsilon}$ for a fixed constant $0 \\leq \\theta < 1$ and any $\\varepsilon>0$. More precisely, letting $\\Lambda^\\sharp$ and $d_k^\\sharp$ be suitable approximants of $\\Lambda$ and $d_k$ and $\\mu^\\sharp = 0$, we show for instance that, for any nilsequence $F(g(n)\\Gamma)$, we have \\[ \\sum_{X < n \\leq X+H} (f(n)-f^\\sharp(n)) F(g(n) \\Gamma) \\ll H \\log^{-A} X \\] when $\\theta = 5/8$ and $f \\in \\{\\Lambda, \\mu, d_k\\}$ or $\\theta = 1/3$ and $f = d_2$. As a consequence, we show that the short interval Gowers norms $\\|f-f^\\sharp\\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ for these choices of $f,\\theta$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals, and show that multiple ergodic averages along primes in short intervals converge in $L^2$. Our innovations include the use of multi-parameter nilsequence equidistribution theorems to control type $II$ sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-07T21:58:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic functions",
      "control type",
      "multi-parameter nilsequence equidistribution theorems",
      "study higher uniformity properties",
      "short interval gowers norms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 86,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}