{
  "id": "1906.02574",
  "version": "v1",
  "published": "2019-06-06T13:28:51.000Z",
  "updated": "2019-06-06T13:28:51.000Z",
  "title": "Nearly $k$-distance sets",
  "authors": [
    "Nóra Frankl",
    "Andrey Kupavskii"
  ],
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "abstract": "We say that a set of points $S\\subset \\mathbb{R}^d$ is an $\\varepsilon$-nearly $k$-distance set if there exist $1\\le t_1\\le \\ldots\\le t_k,$ such that the distance between any two distinct points in $S$ falls into $[t_1,t_1+\\varepsilon]\\cup\\ldots\\cup[t_k,t_k+\\varepsilon]$. In this paper, we study the quantity $M_k(d) = \\lim_{\\varepsilon\\to 0}\\max\\{|S|\\ :\\ S\\text{ is an }\\varepsilon\\text{-nearly } k \\text{-distance set in } \\mathbb{R}^d\\}$ and its relation to the classical quantity $m_k(d)$: the size of the largest $k$-distance set in $\\mathbb{R}^d$. We obtain that $M_k(d) = m_k(d)$ for $k=2,3$, as well as for any fixed $k$, provided that $d$ is sufficiently large. The last result answers a question, proposed by Erd\\H{o}s, Makai and Pach. We also address a closely related Tur\\'an-type problem, studied by Erd\\H{o}s, Makai, Pach, and Spencer in the 80's: given $n$ points in $\\mathbb{R}^d$, how many pairs of them form a distance that belongs to $[t_1,t_1+1]\\cup\\ldots\\cup[t_k,t_k+1],$ where $t_1,\\ldots, t_k$ are fixed and any two points in the set are at distance at least $1$ apart? We establish the connection between this quantity and a quantity closely related to $M_k(d-1)$, as well as obtain an exact answer for the same ranges $k,d$ as above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-06T13:28:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distance set",
      "distinct points",
      "exact answer",
      "closely related turan-type problem",
      "result answers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}