{
  "id": "1510.07179",
  "version": "v1",
  "published": "2015-10-24T19:47:46.000Z",
  "updated": "2015-10-24T19:47:46.000Z",
  "title": "On problems of Danzer and Gowers and dynamics on the space of closed subsets of $\\mathbb{R}^d$",
  "authors": [
    "Omri Solan",
    "Yaar Solomon",
    "Barak Weiss"
  ],
  "categories": [
    "math.DS",
    "math.CO"
  ],
  "abstract": "Considering the space of closed subsets of $\\mathbb{R}^d$, endowed with the Chabauty-Fell topology, and the affine action of $SL_d(\\mathbb{R})\\ltimes\\mathbb{R}^d$, we prove that the only minimal subsystems are the fixed points $\\{\\varnothing\\}$ and $\\{\\mathbb{R}^d\\}$. As a consequence we resolve a question of Gowers concerning the existence of certain Danzer sets: there is no set $Y \\subset \\mathbb{R}^d$ such that for every convex set $\\mathcal{C} \\subset \\mathbb{R}^d$ of volume one, the cardinality of $\\mathcal{C} \\cap Y$ is bounded above and below by nonzero contants independent of $\\mathcal{C}$. We also provide a short independent proof of this fact and deduce a quantitative consequence: for every $\\varepsilon$-net $N$ for convex sets in $[0,1]^d$ there is a convex set of volume $\\varepsilon$ containing at least $\\Omega(\\log\\log(1/\\varepsilon))$ points of $N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-24T19:47:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "closed subsets",
      "convex set",
      "nonzero contants independent",
      "short independent proof",
      "chabauty-fell topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151007179S"
    }
  }
}