{
  "id": "2111.10082",
  "version": "v1",
  "published": "2021-11-19T07:52:36.000Z",
  "updated": "2021-11-19T07:52:36.000Z",
  "title": "On normal numbers and self-similar measures",
  "authors": [
    "Amir Algom",
    "Simon Baker",
    "Pablo Shmerkin"
  ],
  "comment": "This article supersedes arXiv:2107.02699",
  "journal": "Adv. Math. 399 (2022), Paper No. 108276, 17 pp",
  "categories": [
    "math.DS",
    "math.CA",
    "math.NT"
  ],
  "abstract": "Let $\\lbrace f_i(x)=s_i \\cdot x+t_i \\rbrace$ be a self-similar IFS on $\\mathbb{R}$ and let $\\beta >1$ be a Pisot number. We prove that if $\\frac{\\log |s_i|}{\\log \\beta}\\notin \\mathbb{Q}$ for some $i$ then for every $C^1$ diffeomorphism $g$ and every non-atomic self similar measure $\\mu$, the measure $g\\mu$ is supported on numbers that are normal in base $\\beta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-11-19T07:52:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-similar measures",
      "normal numbers",
      "non-atomic self similar measure",
      "pisot number",
      "self-similar ifs"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}