{
  "id": "2211.02873",
  "version": "v1",
  "published": "2022-11-05T10:58:03.000Z",
  "updated": "2022-11-05T10:58:03.000Z",
  "title": "Limit laws in the lattice problem. IV. The special case of $\\mathbb{Z}^{d}$",
  "authors": [
    "Julien Trevisan"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the error of the number of points of the lattice $\\mathbb{Z}^{d}$ that fall into a dilated and translated hypercube centred around $0$ and whose axis are parallel to the axis of coordinates. We show that if $t$, the factor of dilatation, is distributed according to the probability measure $\\frac{1}{T} \\rho(\\frac{t}{T}) dt$ with $\\rho$ being a probability density over $[0,1]$ the error, when normalized by $t^{d-1}$, converges in law when $T \\rightarrow \\infty$ in the case where the translation is of the form $X=(x,\\cdots,x)$ and in the case where the coordinates of $X$ are independent between them, independent from $t$ and distributed according to the uniform law over $[-\\frac{1}{2},\\frac{1}{2}]$. In both cases, we compute the characteristic function of the limit law.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-05T10:58:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "limit law",
      "lattice problem",
      "special case",
      "characteristic function",
      "probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}