{
  "id": "1711.08934",
  "version": "v1",
  "published": "2017-11-24T11:57:50.000Z",
  "updated": "2017-11-24T11:57:50.000Z",
  "title": "Improved bounds for restricted families of projections to planes in $\\mathbb{R}^{3}$",
  "authors": [
    "Tuomas Orponen",
    "Laura Venieri"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "For $e \\in S^{2}$, the unit sphere in $\\mathbb{R}^3$, let $\\pi_{e}$ be the orthogonal projection to $e^{\\perp} \\subset \\mathbb{R}^{3}$, and let $W \\subset \\mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. We prove that if $K \\subset \\mathbb{R}^{3}$ is a Borel set with $\\dim_{\\mathrm{H}} K \\leq \\tfrac{3}{2}$, then $\\dim_{\\mathrm{H}} \\pi_{e}(K) = \\dim_{\\mathrm{H}} K$ for $\\mathcal{H}^{1}$ almost every $e \\in S^{2} \\cap W$, where $\\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure and $\\dim_{\\mathrm{H}}$ the Hausdorff dimension. This was known earlier, due to J\\\"arvenp\\\"a\\\"a, J\\\"arvenp\\\"a\\\"a, Ledrappier and Leikas, for Borel sets $K \\subset \\mathbb{R}^{3}$ with $\\dim_{\\mathrm{H}} K \\leq 1$. We also prove a partial result for sets with dimension exceeding $3/2$, improving earlier bounds by D. Oberlin and R. Oberlin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-24T11:57:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A78"
    ],
    "keywords": [
      "restricted families",
      "borel set",
      "dimensional hausdorff measure",
      "unit sphere",
      "improving earlier bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}