{
  "id": "1811.12559",
  "version": "v1",
  "published": "2018-11-30T00:53:23.000Z",
  "updated": "2018-11-30T00:53:23.000Z",
  "title": "A lower bound for the a.e. behaviour of Hausdorff dimension under vertical projections in the Heisenberg group",
  "authors": [
    "Terence L. J. Harris"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "It is shown that for a Borel set $A$ in the Heisenberg group with $\\dim A >2$, \\[ \\dim P_{\\mathbb{V}^{\\perp}_{\\theta}} A \\geq \\begin{cases} \\frac{\\dim A}{2} &\\text{ if } \\dim A \\in \\left(2, \\frac{5}{2} \\right] \\\\ \\frac{ \\dim A(\\dim A + 2)}{4\\dim A -1} &\\text{ if } \\dim A \\in \\left( \\frac{5}{2} , 4 \\right], \\end{cases} \\] for a.e. $\\theta \\in [0,\\pi)$, where $\\dim$ refers to the Hausdorff dimension under the Kor\\'anyi metric, and $P_{\\mathbb{V}^{\\perp}_{\\theta}}$ is the vertical Heisenberg projection onto the vertical plane at angle $\\theta+ \\frac{\\pi}{2}$. This improves the known lower bounds in the range $2 < \\dim A <\\frac{12+\\sqrt{109}}{7}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-30T00:53:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A80"
    ],
    "keywords": [
      "lower bound",
      "hausdorff dimension",
      "heisenberg group",
      "vertical projections",
      "borel set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}