{
  "id": "1910.01824",
  "version": "v1",
  "published": "2019-10-04T07:35:21.000Z",
  "updated": "2019-10-04T07:35:21.000Z",
  "title": "Mean Value of $S$-arithmetic Siegel transform: Rogers' mean value theorem for $S$-arithmetic Siegel transform and applications to the geometry of numbers",
  "authors": [
    "Jiyoung Han"
  ],
  "comment": "27 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "We prove the second moment theorem for Siegel transform defined over the space of unimodular $S$-lattices in $\\mathbb Q_S^d$, $d\\ge 3$, following the work of Rogers (1955). As applications, we obtain the random statements of Gauss circle problem for any convex sets in $\\mathbb Q_S^d$ containing the origin and of the effective Oppenheim conjecture for $S$-arithmetic quadratic forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-04T07:35:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11H60",
      "11P21",
      "37A45"
    ],
    "keywords": [
      "arithmetic siegel transform",
      "mean value theorem",
      "applications",
      "second moment theorem",
      "gauss circle problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}