{
  "id": "1905.06483",
  "version": "v1",
  "published": "2019-05-16T00:43:09.000Z",
  "updated": "2019-05-16T00:43:09.000Z",
  "title": "Occurrence of distances in vector spaces over prime fields",
  "authors": [
    "Thang Pham",
    "Le Anh Vinh"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $\\mathbb{F}_q$ be an arbitrary finite field, and $\\mathcal{E}$ be a set of points in $\\mathbb{F}_q^d$. Let $\\Delta(\\mathcal{E})$ be the set of distances determined by pairs of points in $\\mathcal{E}$. By using the Kloosterman sums, Iosevich and Rudnev proved that if $|\\mathcal{E}|\\ge 4q^{\\frac{d+1}{2}}$, then $\\Delta(\\mathcal{E})=\\mathbb{F}_q$. In general, this result is sharp in odd dimensional spaces over arbitrary finite fields. In this paper, we use the recent point-plane incidence bound due to Rudnev to prove that if $\\mathcal{E}$ has Cartesian product structure in vector spaces over prime fields, then we can break the exponent $(d+1)/2$, and still cover \\textbf{all distances}. We also show that the number of pairs of points in $\\mathcal{E}$ of any given distance is close to its expected value.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-16T00:43:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "prime fields",
      "vector spaces",
      "arbitrary finite field",
      "occurrence",
      "cartesian product structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}