{
  "id": "1811.04450",
  "version": "v1",
  "published": "2018-11-11T18:58:50.000Z",
  "updated": "2018-11-11T18:58:50.000Z",
  "title": "Borel complexity of sets of normal numbers via generic points in subshifts with specification",
  "authors": [
    "Dylan Airey",
    "Steve Jackson",
    "Dominik Kwietniak",
    "Bill Mance"
  ],
  "comment": "A talk explaining this paper may be found at https://www.youtube.com/watch?v=g9va0ZzVIjA",
  "categories": [
    "math.DS",
    "math.LO",
    "math.NT"
  ],
  "abstract": "We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base $r$ expansions, and their various generalisations: generalised L\\\"uroth series expansions and $\\beta$-expansions. In fact, we consider subshifts over a countable alphabet generated by all possible expansions of numbers in $[0,1)$. Then normal numbers correspond to generic points of shift-invariant measures. It turns out that for these subshifts the set of generic points for a shift-invariant probability measure is precisely at the third level of the Borel hierarchy (it is a $\\Pi^0_3$-complete set, meaning that it is a countable intersection of $F_\\sigma$-sets, but it is not possible to write it as a countable union of $G_\\delta$-sets). We also solve a problem of Sharkovsky--Sivak on the Borel complexity of the basin of statistical attraction. The crucial dynamical feature we need is a feeble form of specification. All expansions named above generate subshifts with this property. Hence the sets of normal numbers under consideration are $\\Pi^0_3$-complete.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-11T18:58:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "borel complexity",
      "generic points",
      "specification",
      "shift-invariant probability measure",
      "normal numbers correspond"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}