{
  "id": "2204.08428",
  "version": "v1",
  "published": "2022-04-15T15:50:36.000Z",
  "updated": "2022-04-15T15:50:36.000Z",
  "title": "Thickness and a gap lemma in $\\mathbb{R}^d$",
  "authors": [
    "Alexia Yavicoli"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "We give a definition of thickness in $\\mathbb{R}^d$ that is useful even for totally disconnected sets, and prove a Gap Lemma type result. We also guarantee an interval of distances in any direction in thick compact sets, relate thick sets (for this definition of thickness) with winning sets, give a lower bound for the Hausdorff dimension of the intersection of countably many of them, a result guaranteeing the presence of large patterns, and lower bounds for the Hausdorff dimension of a set in relationship with its thickness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-15T15:50:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "11B25",
      "28A78"
    ],
    "keywords": [
      "hausdorff dimension",
      "lower bound",
      "gap lemma type result",
      "thick compact sets",
      "relate thick sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}