{
  "id": "2211.00412",
  "version": "v1",
  "published": "2022-11-01T12:11:31.000Z",
  "updated": "2022-11-01T12:11:31.000Z",
  "title": "Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$",
  "authors": [
    "Stephan Baier"
  ],
  "comment": "29 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Friedlander and Iwaniec investigated integral solutions $(x_1,x_2,x_3)$ of the equation $x_1^2+x_2^2-x_3^2=D$, where $D$ is square-free and satisfies the congruence condition $D\\equiv 5\\bmod{8}$. They obtained an asymptotic formula for solutions with $x_3\\asymp M$, where $M$ is much smaller than $\\sqrt{D}$. To be precise, their condition is $M\\ge D^{1/2-1/1332}$. Their analysis led them to averages of certain Weyl sums. The condition of $D$ being square-free is essential in their work. We investigate the \"opposite\" case when $D=n^2$ is a square of an odd integer $n$. This case is different in nature and leads to sums of Kloosterman sums. We obtain an asymptotic formula for solutions with $x_3\\asymp M$, where $M\\ge D^{1/2}\\exp\\left(-(\\log D)^{1/2-\\varepsilon}\\right)$ for any fixed $\\varepsilon>0$. It is a certain term not containing Kloosterman sums that prevents us from saving a power of $D$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-01T12:11:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E20",
      "11L05",
      "11D09"
    ],
    "keywords": [
      "asymptotic formula",
      "congruence condition",
      "integral solutions",
      "weyl sums",
      "odd integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}