{
  "id": "math/0406375",
  "version": "v1",
  "published": "2004-06-18T17:24:45.000Z",
  "updated": "2004-06-18T17:24:45.000Z",
  "title": "The sharp Hausdorff measure condition for length of projections",
  "authors": [
    "Yuval Peres",
    "Boris Solomyak"
  ],
  "comment": "11 pages, 1 figure",
  "journal": "Proc. Amer. Math. Soc. 133 (2005), no. 11, 3371--3379",
  "categories": [
    "math.CA"
  ],
  "abstract": "In a recent paper, Pertti Mattila asked which gauge functions $\\phi$ have the property that for any planar Borel set $A$ with positive Hausdorff measure in gauge $\\phi$, the projection of $A$ to almost every line has positive length. We show that integrability near zero of $\\phi(r)/(r^2)$, which is known to be sufficient for this property, is also necessary if $\\phi$ is regularly varying. Our proof is based on a random construction adapted to the gauge function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-06-18T17:24:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A75",
      "60D05",
      "28A78"
    ],
    "keywords": [
      "sharp hausdorff measure condition",
      "projection",
      "gauge function",
      "planar borel set",
      "pertti mattila"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6375P"
    }
  }
}