{
  "id": "1608.06398",
  "version": "v1",
  "published": "2016-08-23T06:46:38.000Z",
  "updated": "2016-08-23T06:46:38.000Z",
  "title": "An improvement on the number of simplices in $\\mathbb{F}_q^d$",
  "authors": [
    "Thang Pham",
    "Duc Hiep Pham",
    "Anh Vinh Le"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{E}$ be a set of points in $\\mathbb{F}_q^d$. Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2016) proved that if $|\\mathcal{E}|\\gg q^{d-\\frac{d-1}{k+1}}$ then $\\mathcal{E}$ determines a positive proportion of all $k$-simplices. In this paper, we give an improvement of this result in the case when $\\mathcal{E}$ is the Cartesian product of sets. More precisely, we show that if $\\mathcal{E}$ is the Cartesian product of sets and $q^{\\frac{kd}{k+1-1/d}}=o(|\\mathcal{E}|)$, the number of congruence classes of $k$-simplices determined by $\\mathcal{E}$ is at least $(1-o(1))q^{\\binom{k+1}{2}}$, and in some cases our result is sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-23T06:46:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "improvement",
      "cartesian product",
      "congruence classes",
      "determines",
      "pakianathan"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}