{
  "id": "2212.02984",
  "version": "v1",
  "published": "2022-12-05T16:34:41.000Z",
  "updated": "2022-12-05T16:34:41.000Z",
  "title": "Cantor sets with high-dimensional projections",
  "authors": [
    "Olga Frolkina"
  ],
  "journal": "Topol. Appl. 275 (2020) 107020",
  "doi": "10.1016/j.topol.2019.107020",
  "categories": [
    "math.GT",
    "math.GN"
  ],
  "abstract": "In 1994, J.Cobb constructed a tame Cantor set in $\\mathbb R^3$ each of whose projections into $2$-planes is one-dimensional. We show that an Antoine's necklace can serve as an example of a Cantor set all of whose projections are one-dimensional and connected. We prove that each Cantor set in $\\mathbb R^n$, $n\\geqslant 3$, can be moved by a small ambient isotopy so that the projection of the resulting Cantor set into each $(n-1)$-plane is $(n-2)$-dimensional. We show that if $X\\subset \\mathbb R^n$, $n\\geqslant 2$, is a zero-dimensional compactum whose projection into some plane $\\Pi\\subset \\mathbb R^n$ with $\\dim \\Pi \\in \\{1, 2, n-2, n-1\\}$ is zero-dimensional, then $X$ is tame; this extends some particular cases of the results of D.R.McMillan, Jr. (1964) and D.G.Wright, J.J.Walsh (1982). We use the technique of defining sequences which comes back to Louis Antoine.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-05T16:34:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54F45",
      "57N15",
      "57M30"
    ],
    "keywords": [
      "high-dimensional projections",
      "tame cantor set",
      "small ambient isotopy",
      "one-dimensional",
      "antoines necklace"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}