{
  "id": "1409.8070",
  "version": "v1",
  "published": "2014-09-29T10:59:33.000Z",
  "updated": "2014-09-29T10:59:33.000Z",
  "title": "Codimension formulae for the intersection of fractal subsets of Cantor spaces",
  "authors": [
    "Casey Donoven",
    "Kenneth Falconer"
  ],
  "categories": [
    "math.MG"
  ],
  "abstract": "We examine the dimensions of the intersection of a subset $E$ of an $m$-ary Cantor space $\\mathcal{C}^m$ with the image of a subset $F$ under a random isometry with respect to a natural metric. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. The dimensions of the intersections are typically $\\max\\{\\dim E +\\dim F -\\dim \\mathcal{C}^m, 0\\}$, akin to other codimension theorems. The upper estimates come from the expected sizes of coverings, whilst the lower estimate is more intricate, using martingales to define a random measure on the intersection to facilitate a potential theoretic argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-29T10:59:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "20E08",
      "60G57"
    ],
    "keywords": [
      "intersection",
      "codimension formulae",
      "fractal subsets",
      "ary cantor space",
      "potential theoretic argument"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.8070D"
    }
  }
}