{
  "id": "2010.01096",
  "version": "v1",
  "published": "2020-10-02T16:50:07.000Z",
  "updated": "2020-10-02T16:50:07.000Z",
  "title": "On the Distribution of the Number of Lattice Points in Norm Balls on the Heisenberg Groups",
  "authors": [
    "Yoav A. Gath"
  ],
  "comment": "33 pages, comments are welcome",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "We investigate the fluctuations in the number of integral lattice points on the Heisenberg groups which lie inside a Cygan-Kor{\\'a}nyi norm ball of large radius. Let $\\mathcal{E}_{q}(x)=\\big|\\mathbb{Z}^{2q+1}\\cap\\delta_{x}\\mathcal{B}\\big|-\\textit{vol}\\big(\\mathcal{B}\\big)x^{2q+2}$ denote the error term which occurs for this lattice point counting problem on the Heisenberg group $\\mathbb{H}_{q}$, where $\\mathcal{B}$ is the unit ball in the Cygan-Kor{\\'a}nyi norm and $\\delta_{x}$ is the Heisenberg-dilation by $x>0$. For $q\\geq3$ we consider the suitably normalized error term $\\mathcal{E}_{q}(x)/x^{2q-1}$, and prove it has a limiting value distribution which is absolutely continuous with respect to the Lebesgue measure. We show that the defining density for this distribution, denoted by $\\mathcal{P}_{q}(\\alpha)$, can be extended to the whole complex plane $\\mathbb{C}$ as an entire function of $\\alpha$ and satisfies for any non-negative integer $j\\geq0$ and any $\\alpha\\in\\mathbb{R}$, $|\\alpha|>\\alpha_{q,j}$, the bound: \\begin{equation*} \\begin{split} \\big|\\mathcal{P}^{(j)}_{q}(\\alpha)\\big|\\leq\\exp{\\Big(-|\\alpha|^{4-\\beta/\\log\\log{|\\alpha|}}\\Big)} {split} {equation*} where $\\beta>0$ is an absolute constant. In addition, we give an explicit formula for the $j$-th integral moment of the density $\\mathcal{P}_{q}(\\alpha)$ for any integer $j\\geq1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-02T16:50:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heisenberg group",
      "distribution",
      "lattice point counting problem",
      "integral lattice points",
      "nyi norm ball"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}