{
  "id": "1702.06785",
  "version": "v1",
  "published": "2017-02-22T13:12:11.000Z",
  "updated": "2017-02-22T13:12:11.000Z",
  "title": "Singularity versus exact overlaps for self-similar measures",
  "authors": [
    "Károly Simon",
    "Lajos Vágó"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this note we present some one-parameter families of homogeneous self-similar measures on the line such that - the similarity dimension is greater than $1$ for all parameters and - the singularity of some of the self-similar measures from this family is not caused by exact overlaps between the cylinders. We can obtain such a family as the angle-$\\alpha$ projections of the natural measure of the Sierpi\\'nski carpet. We present more general one-parameter families of self-similar measures $\\nu_\\alpha$, such that the set of parameters $\\alpha$ for which $\\nu_\\alpha$ is singular is a dense $G_\\delta$ set but this \"exceptional\" set of parameters of singularity has zero Hausdorff dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-22T13:12:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A99"
    ],
    "keywords": [
      "exact overlaps",
      "singularity",
      "zero hausdorff dimension",
      "general one-parameter families",
      "parameters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}