{
  "id": "2111.06139",
  "version": "v1",
  "published": "2021-11-11T10:46:46.000Z",
  "updated": "2021-11-11T10:46:46.000Z",
  "title": "Asymptotic distribution for pairs of linear and quadratic forms at integral vectors",
  "authors": [
    "Jiyoung Han",
    "Seonhee Lim",
    "Keivan Mallahi-Karai"
  ],
  "comment": "23 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "We study the joint distribution of values of a pair consisting of a quadratic form $q$ and a linear form $\\mathbf l$ over the set of integral vectors, a problem initiated by Dani-Margulis (1989). In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if $n \\ge 5$ then under the assumptions that for every $(\\alpha, \\beta ) \\in \\mathbb R^2 \\setminus \\{ (0,0) \\}$, the form $\\alpha q + \\beta \\mathbf l^2$ is irrational and that the signature of the restriction of $q$ to the kernel of $\\mathbf l$ is $(p, n-1-p)$, where $3\\le p \\le n-2$, the number of vectors $v \\in \\mathbb Z^n$ for which $\\|v\\| < T$, $a < q(v) < b$ and $c< \\mathbf l(v) < d$ is asymptotically $$ C(q, \\mathbf l)(d-c)(b-a)T^{n-3} , $$ as $T \\to \\infty$, where $C(q, \\mathbf l)$ only depends on $q$ and $\\mathbf l$. The density of the set of joint values of $(q, \\mathbf l)$ under the same assumptions is shown by Gorodnik (2004).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-11-11T10:46:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B15"
    ],
    "keywords": [
      "quadratic form",
      "integral vectors",
      "asymptotic distribution",
      "joint distribution",
      "joint values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}