{
  "id": "2310.07538",
  "version": "v1",
  "published": "2023-10-11T14:41:20.000Z",
  "updated": "2023-10-11T14:41:20.000Z",
  "title": "Hausdorff dimension of plane sections and general intersections",
  "authors": [
    "Pertti Mattila"
  ],
  "comment": "10 pages. arXiv admin note: substantial text overlap with arXiv:2005.11790",
  "categories": [
    "math.CA"
  ],
  "abstract": "This paper extends some results of [M5] and [M3], in particular, removing assumptions of positive lower density. We give conditions on a general family $P_{\\lambda}:\\mathbb{R}^{n}\\to\\mathbb{R}^{m}, \\lambda \\in \\Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\\dim A\\cap P_{\\lambda}^{-1}\\{u\\}=s-m$ holds generically for measurable sets $A\\subset\\mathbb{R}^{n}$ with positive and finite $s$-dimensional Hausdorff measure, $s>m$. As an application we prove for measurable sets $A,B\\subset\\mathbb{R}^{n}$ with positive $s$- and $t$-dimensional measures that if $s + (n-1)t/n > n$, then $\\dim A\\cap (g(B)+z) \\geq s+t - n$ for almost all rotations $g$ and for positively many $z\\in\\mathbb{R}^{n}$. We shall also give an application on the estimates of the dimension of the set of exceptional rotations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-11T14:41:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "general intersections",
      "plane sections",
      "dimensional hausdorff measure",
      "measurable sets",
      "hausdorff dimension formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}