{
  "id": "1403.6138",
  "version": "v2",
  "published": "2014-03-24T20:22:33.000Z",
  "updated": "2015-02-04T22:15:40.000Z",
  "title": "On the sums of any k points in finite fields",
  "authors": [
    "David Covert",
    "Doowon Koh",
    "Youngjin Pi"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO",
    "math.CA",
    "math.NT"
  ],
  "abstract": "For a set $E\\subset \\mathbb F_q^d$, we define the $k$-resultant magnitude set as $ \\Delta_k(E) =\\{\\|\\textbf{x}_1 + \\dots + \\textbf{x}_k\\|\\in \\mathbb F_q: \\textbf{x}_1, \\dots, \\textbf{x}_k \\in E\\},$ where $\\|\\textbf{v}\\|=v_1^2+\\cdots+ v_d^2$ for $\\textbf{v}=(v_1, \\ldots, v_d) \\in \\mathbb F_q^d.$ In this paper we find a connection between a lower bound of the cardinality of the $k$-resultant magnitude set and the restriction theorem for spheres in finite fields. As a consequence, it is shown that if $E\\subset \\mathbb F_q^d$ with $|E|\\geq C q^{\\frac{d+1}{2}-\\frac{1}{6d+2}},$ then $|\\Delta_3(E)|\\geq c q$ for $d = 4$ or $d = 6$, and $|\\Delta_4(E)| \\geq cq$ for even dimensions $d \\geq 8.$ In addition, we prove that if $d\\geq 8$ is even, and $|E|\\geq C_\\varepsilon ~q^{\\frac{d+1}{2} - \\frac{1}{9d -18} + \\varepsilon}$ for $\\varepsilon >0$, then $|\\Delta_3(E)|\\geq c q.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-24T20:22:33.000Z",
      "abstract": "For a set $E\\subset \\mathbb F_q^d$, we define the $k$-resultant magnitude set as $ \\Delta_k(E) =\\{\\|x^1 + \\dots + x^k\\|\\in \\mathbb F_q: x^1, \\dots, x^k \\in E\\},$ where $\\|\\alpha\\|=\\alpha_1^2+\\cdots+ \\alpha_d^2$ for $\\alpha=(\\alpha_1, \\ldots, \\alpha_d) \\in \\mathbb F_q^d.$ In this paper we find a connection between a lower bound of the cardinality of the $k$-resultant magnitude set and the restriction theorem for spheres in finite fields. As a consequence, it is shown that if $E\\subset \\mathbb F_q^d$ with $|E|\\geq C q^{\\frac{d+1}{2}-\\frac{1}{6d+2}},$ then $|\\Delta_3(E)|\\geq c q$ for $d = 4$ or $d = 6$, and $|\\Delta_k(E)| \\geq cq$ for even dimensions $d \\geq 4$ and $k \\geq 4$. In addition, we prove that if $d\\geq 8$ is even, and $|E|\\geq C_\\varepsilon ~q^{\\frac{d+1}{2}-\\frac{1}{9d-18}+\\varepsilon}$ for $\\varepsilon >0$, then $|\\Delta_3(E)|\\geq c q$ for some $0<c<1.$",
      "comment": "14 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-04T22:15:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10",
      "42B05",
      "11T23"
    ],
    "keywords": [
      "finite fields",
      "resultant magnitude set",
      "lower bound",
      "restriction theorem",
      "connection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.6138C"
    }
  }
}