{
  "id": "1812.11556",
  "version": "v1",
  "published": "2018-12-30T15:30:43.000Z",
  "updated": "2018-12-30T15:30:43.000Z",
  "title": "On the structure of distance sets over prime fields",
  "authors": [
    "Thang Pham",
    "Andrew Suk"
  ],
  "comment": "8 pages. Submitted for publication",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $\\mathbb{F}_q$ be a finite field of order $q$ and $\\mathcal{E}$ be a set in $\\mathbb{F}_q^d$. The distance set of $\\mathcal{E}$, denoted by $\\Delta(\\mathcal{E})$, is the set of distinct distances determined by the pairs of points in $\\mathcal{E}$. Very recently, Iosevich, Koh, and Parshall (2018) proved that if $|\\mathcal{E}|\\gg q^{d/2}$, then the quotient set of $\\Delta(\\mathcal{E})$ satisfies \\[\\left\\vert\\frac{\\Delta(\\mathcal{E})}{\\Delta(\\mathcal{E})}\\right\\vert=\\left\\vert \\left\\lbrace\\frac{a}{b}\\colon a, b\\in \\Delta(\\mathcal{E}), b\\ne 0\\right\\rbrace\\right\\vert\\gg q.\\] In this paper, we break the exponent $d/2$ when $\\mathcal{E}$ is a Cartesian product of sets over a prime field. More precisely, let $p$ be a prime and $A\\subset \\mathbb{F}_p$. If $\\mathcal{E}=A^d\\subset \\mathbb{F}_p^d$ and $|\\mathcal{E}|\\gg p^{\\frac{d}{2}-\\varepsilon}$ for some $\\varepsilon>0$, then we have \\[\\left\\vert\\frac{\\Delta(\\mathcal{E})}{\\Delta(\\mathcal{E})}\\right\\vert, ~\\left\\vert \\Delta(\\mathcal{E})\\cdot \\Delta(\\mathcal{E})\\right\\vert \\gg p.\\] Such improvements are not possible over arbitrary finite fields. These results give us a better understanding about the structure of distance sets and the Erd\\H{o}s-Falconer distance conjecture over finite fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-30T15:30:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distance set",
      "prime field",
      "arbitrary finite fields",
      "distinct distances",
      "distance conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}