{
  "id": "2006.15569",
  "version": "v1",
  "published": "2020-06-28T10:59:39.000Z",
  "updated": "2020-06-28T10:59:39.000Z",
  "title": "On the packing dimension of Furstenberg sets",
  "authors": [
    "Pablo Shmerkin"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "We prove that if $\\alpha\\in (0,1/2]$, then the packing dimension of a set $E\\subset\\mathbb{R}^2$ for which there exists a set of lines of dimension $1$ intersecting $E$ in dimension $\\ge \\alpha$ is at least $1/2+\\alpha+c(\\alpha)$ for some $c(\\alpha)>0$. In particular, this holds for $\\alpha$-Furstenberg sets, that is, sets having intersection of Hausdorff dimension $\\ge\\alpha$ with at least one line in every direction. Together with an earlier result of T. Orponen, this provides an improvement for the packing dimension of $\\alpha$-Furstenberg sets over the \"trivial\" estimate for all values of $\\alpha\\in (0,1)$. The proof extends to more general families of lines, and shows that the scales at which an $\\alpha$-Furstenberg set resembles a set of dimension close to $1/2+\\alpha$, if they exist, are rather sparse.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-28T10:59:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A80"
    ],
    "keywords": [
      "packing dimension",
      "furstenberg set resembles",
      "earlier result",
      "proof extends",
      "general families"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}