{
  "id": "1912.01523",
  "version": "v1",
  "published": "2019-12-03T16:55:08.000Z",
  "updated": "2019-12-03T16:55:08.000Z",
  "title": "On sets containing a unit distance in every direction",
  "authors": [
    "Pablo Shmerkin",
    "Han Yu"
  ],
  "comment": "13 pages, 2 figures",
  "categories": [
    "math.CA",
    "math.CO",
    "math.MG"
  ],
  "abstract": "We investigate the box dimensions of compact sets in $\\mathbb{R}^n$ that contain a unit distance in every direction (such sets may have zero Hausdorff dimension). Among other results, we show that the lower box dimension must be at least $\\frac{n^2(n-1)}{2n^2-1}$ and can be at most $\\frac{n(n-1)}{2n-1}$. This quantifies in a certain sense how far the unit sphere $S^{n-1}$ is from being a difference set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-03T16:55:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D99",
      "28A80"
    ],
    "keywords": [
      "unit distance",
      "sets containing",
      "lower box dimension",
      "zero hausdorff dimension",
      "compact sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}