{
  "id": "2112.09044",
  "version": "v2",
  "published": "2021-12-16T17:35:05.000Z",
  "updated": "2022-01-19T01:59:42.000Z",
  "title": "On the distance sets spanned by sets of dimension $d/2$ in $\\mathbb{R}^d$",
  "authors": [
    "Pablo Shmerkin",
    "Hong Wang"
  ],
  "comment": "v2: added figures. 71 pages, 4 figures",
  "categories": [
    "math.CA",
    "math.CO",
    "math.MG"
  ],
  "abstract": "We establish the dimension version of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In dimensions $d=2$ or $3$, we obtain the first explicit estimates for the dimensions of distance sets of general Borel sets of dimension $d/2$; for example, we show that the set of distances spanned by a planar Borel set of Hausdorff dimension $1$ has Hausdorff dimension at least $(\\sqrt{5}-1)/2\\approx 0.618$. In higher dimensions we obtain explicit estimates for the lower Minkowski dimension of the distance sets of sets of dimension $d/2$. These results rely on new estimates for the dimensions of radial projections that may have independent interest.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-19T01:59:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A80"
    ],
    "keywords": [
      "hausdorff dimension",
      "falconers distance set conjecture",
      "general borel sets",
      "lower minkowski dimension",
      "planar borel set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 71,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}