{
  "id": "1302.5792",
  "version": "v3",
  "published": "2013-02-23T12:42:35.000Z",
  "updated": "2014-12-08T07:20:27.000Z",
  "title": "Equidistribution from Fractals",
  "authors": [
    "Michael Hochman",
    "Pablo Shmerkin"
  ],
  "comment": "46 pages. v3: minor corrections and elaborations",
  "categories": [
    "math.DS",
    "math.CA",
    "math.NT"
  ],
  "abstract": "We give a fractal-geometric condition for a measure on [0,1] to be supported on points x that are normal in base n, i.e. such that the sequence x,nx,n^2 x,... equidistributes modulo 1. This condition is robust under C^1 coordinate changes, and it applies also when n is a Pisot number and equidistribution is understood with respect to the beta-map and Parry measure. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent integers and proving a Rudolph-Johnson-type theorem for certain pairs of beta transformations.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-07T11:09:03.000Z",
      "comment": "44 pages. v2: corrected statement and more detail in Lemma 4.16 and Corollary 4.17",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-12-08T07:20:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K16",
      "11A63",
      "28A80",
      "28D05"
    ],
    "keywords": [
      "equidistribution",
      "beta transformations",
      "equidistributes modulo",
      "coordinate changes",
      "pisot number"
    ],
    "publication": {
      "doi": "10.1007/s00222-014-0573-5",
      "journal": "Inventiones Mathematicae",
      "year": 2015,
      "month": "Oct",
      "volume": 202,
      "number": 1,
      "pages": 427
    },
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015InMat.202..427H"
    }
  }
}