{
  "id": "2208.06896",
  "version": "v1",
  "published": "2022-08-14T18:51:01.000Z",
  "updated": "2022-08-14T18:51:01.000Z",
  "title": "Length of sets under restricted families of projections onto lines",
  "authors": [
    "Terence L. J. Harris"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "Let $\\gamma: I \\to S^2$ be a $C^2$ curve with $\\det(\\gamma, \\gamma', \\gamma'')$ nonvanishing, and for each $\\theta \\in I$ let $\\rho_{\\theta}$ be orthogonal projection onto the line through $\\gamma(\\theta)$. It is shown that if $A \\subseteq \\mathbb{R}^3$ is a Borel set of Hausdorff dimension strictly greater than 1, then $\\rho_{\\theta}(A)$ has positive length for a.e. $\\theta \\in I$. This answers a question raised by K\\\"aenm\\\"aki, Orponen and Venieri.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-14T18:51:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "28A80"
    ],
    "keywords": [
      "restricted families",
      "hausdorff dimension strictly greater",
      "orthogonal projection",
      "borel set",
      "positive length"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}