{
  "id": "1312.2812",
  "version": "v1",
  "published": "2013-12-10T14:31:22.000Z",
  "updated": "2013-12-10T14:31:22.000Z",
  "title": "Measure and Hausdorff dimension of randomized Weierstrass-type functions",
  "authors": [
    "Julia Romanowska"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we consider functions of the type $$f(x) = \\sum_{n=0}^\\infty a_n g(b_nx+\\theta_n),$$ where $(a_n)$ are independent random variables uniformly distributed on $(-a^n, a^n)$ for some $0<a<1$, $b_{n+1}/b_n \\geq b >1$, $a^2b> 1$ and $g$ is a $C^1$ periodic real function with finite number of critical points in every bounded interval. We prove that the occupation measure for $f$ has $L^2$ density almost surely. Furthermore, the Hausdorff dimension of the graph of $f$ is almost surely equal to $D = 2+ \\log{a}/\\log{b}$ provided $ b = \\lim_{n\\rightarrow \\infty}b_{n+1}/b_n>1$ and $ab>1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-10T14:31:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A78",
      "37A45"
    ],
    "keywords": [
      "randomized weierstrass-type functions",
      "hausdorff dimension",
      "periodic real function",
      "independent random variables",
      "occupation measure"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/0951-7715/27/4/787",
      "journal": "Nonlinearity",
      "year": 2014,
      "month": "Apr",
      "volume": 27,
      "number": 4,
      "pages": 787
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014Nonli..27..787R"
    }
  }
}