{
  "id": "1609.02131",
  "version": "v1",
  "published": "2016-09-07T19:34:13.000Z",
  "updated": "2016-09-07T19:34:13.000Z",
  "title": "A new lower bound for Garsia's entropy of Bernoulli convolutions",
  "authors": [
    "Kevin G. Hare",
    "Nikita Sidorov"
  ],
  "comment": "8 pages, no figures",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "Let $\\beta\\in(1,2)$ and let $H_\\beta$ denote Garsia's entropy for the Bernoulli convolution $\\mu_\\beta$ associated with $\\beta$. In the present paper we show that $H_\\beta>0.82$ for all $\\beta \\in (1, 2)$ and improve this bound for certain ranges. Combined with a recent result by Hochman, this yields $\\dim (\\mu_\\beta)>0.82$ for all algebraic $\\beta$. In addition, we show that if an algebraic $\\beta$ is such that $[\\mathbb{Q}(\\beta): \\mathbb{Q}(\\beta^k)] = k$ for some $k \\geq 2$, then $\\dim(\\mu_\\beta)=1$. Such is, for instance, any root of a Pisot number which is not a Pisot number itself.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-07T19:34:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A30",
      "11R06"
    ],
    "keywords": [
      "bernoulli convolution",
      "lower bound",
      "pisot number",
      "denote garsias entropy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}