{
  "id": "1605.02490",
  "version": "v1",
  "published": "2016-05-09T09:26:47.000Z",
  "updated": "2016-05-09T09:26:47.000Z",
  "title": "Asymptotic distribution of values of isotropic quadratic forms at $S$-integral points",
  "authors": [
    "Jiyoung Han",
    "Seonhee Lim",
    "Keivan Mallahi-Karai"
  ],
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "We prove a generalization of a theorem of Eskin-Margulis-Mozes in an $S$-arithmetic setup: suppose that we are given a finite set of places $S$ over $\\mathbb{Q}$ containing the archimedean place, an irrational isotropic form ${\\mathbf q}$ of rank $n\\geq 4$ on $\\mathbb{Q}_S$, a product of $p$-adic intervals ${\\mathbb{I}}_p$, and a product $\\Omega$ of star-shaped sets. We show that if the real part of ${\\mathbf q}$ is not of signature $(2,1)$ or $(2,2)$, then the number of $S$-integral vectors ${\\mathbf v} \\in {\\mathsf{T}} \\Omega$ satisfying simultaneously ${\\mathbf q}({\\mathbf v}) \\in {\\mathbb{I}}_p$ for $p \\in S$ is asymptotically $ \\lambda({\\mathbf q}, \\Omega) |{\\mathbb{I}}| \\cdot \\| {\\mathsf{T}} \\|^{n-2}$, as ${\\mathsf{T}}$ goes to infinity, where $|{\\mathbb{I}}|$ is the product of Haar measures of the $p$-adic intervals ${\\mathbb{I}}_p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-09T09:26:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "isotropic quadratic forms",
      "integral points",
      "asymptotic distribution",
      "adic intervals",
      "irrational isotropic form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}