{
  "id": "2102.13059",
  "version": "v1",
  "published": "2021-02-25T18:10:18.000Z",
  "updated": "2021-02-25T18:10:18.000Z",
  "title": "The range of dimensions of microsets",
  "authors": [
    "Richárd Balka",
    "Márton Elekes",
    "Viktor Kiss"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CA",
    "math.PR"
  ],
  "abstract": "We say that $E$ is a microset of the compact set $K\\subset \\mathbb{R}^d$ if there exist sequences $\\lambda_n\\geq 1$ and $u_n\\in \\mathbb{R}^d$ such that $(\\lambda_n K + u_n ) \\cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and moreover, $E \\cap (0, 1)^d \\neq \\emptyset$. The main result of the paper is that for a non-empty set $A\\subset [0,d]$ there is a compact set $K\\subset \\mathbb{R}^d$ such that the set of Hausdorff dimensions attained by the microsets of $K$ equals $A$ if and only if $A$ is analytic and contains its infimum and supremum. This answers another question of Fraser, Howroyd, K\\\"aenm\\\"aki, and Yu. We show that for every compact set $K\\subset \\mathbb{R}^d$ and non-empty analytic set $A\\subset [0,\\dim_H K]$ there is a set $\\mathcal{C}$ of compact subsets of $K$ which is compact in the Hausdorff metric and $\\{\\dim_H C: C\\in \\mathcal{C} \\}=A$. The proof relies on the technique of stochastic co-dimension applied for a suitable coupling of fractal percolations with generation dependent retention probabilities. We also examine the analogous problems for packing and box dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-25T18:10:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A80",
      "28A05",
      "82B43"
    ],
    "keywords": [
      "compact set",
      "generation dependent retention probabilities",
      "hausdorff metric",
      "non-empty analytic set",
      "main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}