{
  "id": "2007.00429",
  "version": "v1",
  "published": "2020-07-01T12:36:20.000Z",
  "updated": "2020-07-01T12:36:20.000Z",
  "title": "An upper bound for the size of $s$-distance sets in real algebraic sets",
  "authors": [
    "Gábor Hegedüs",
    "Lajos Rónyai"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO",
    "math.AC",
    "math.MG"
  ],
  "abstract": "In a recent paper Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester's Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of $s$-distance sets (subsets $\\mbox{$\\cal A$}\\subseteq {\\mathbb R}^n$ which determine at most $s$ different distances). In this paper we extend their work and prove upper bounds for the size of $s$-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-Goethals-Seidel on spherical $s$-distance sets and a generalization of a bound by Bannai-Kawasaki-Nitamizu-Sato on $s$-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Gr\\\"obner basis techniques.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-01T12:36:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C45",
      "13P10",
      "05D99"
    ],
    "keywords": [
      "distance sets",
      "real algebraic sets",
      "upper bound",
      "real quadratic forms",
      "simple proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}