{
  "id": "1610.03804",
  "version": "v1",
  "published": "2016-10-12T17:51:34.000Z",
  "updated": "2016-10-12T17:51:34.000Z",
  "title": "Small Sets containing any Pattern",
  "authors": [
    "Ursula Molter",
    "Alexia Yavicoli"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Given a dimension function $h$, and a family of functions $\\mathcal{F}$ that satisfy certain conditions, we construct a closed set $E$ in $\\mathbb{R}^N$, of $h$-Hausdorff measure zero, such that for any finite set $\\{ f_1,\\ldots,f_n\\}\\subseteq \\mathcal{F}$, $E$ satisfies that $\\bigcap_{i=1}^n f_i(E)\\neq\\emptyset$. We obtain an analogous result for the preimages, from which we are able to prove that there exists a closed set in $\\mathbb{R}$, of zero $h$-Hausdorff measure, that contains any polynomial pattern. The set $E$ can - in some cases - chosen to be perfect. Additionally we prove some related result for countable intersections, obtaining, instead of a closed set, an $\\mathcal{F}_{\\sigma}$ set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-12T17:51:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A71",
      "10L20"
    ],
    "keywords": [
      "small sets containing",
      "closed set",
      "hausdorff measure zero",
      "finite set",
      "dimension function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}