{
  "id": "1910.11213",
  "version": "v1",
  "published": "2019-10-24T15:18:42.000Z",
  "updated": "2019-10-24T15:18:42.000Z",
  "title": "Turing Degrees and Randomness for Continuous Measures",
  "authors": [
    "Mingyang Li",
    "Jan Reimann"
  ],
  "comment": "22 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the \"dissipation\" function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every $\\Delta^0_2$-degree contains an NCR element.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-24T15:18:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03D32",
      "03D25",
      "03D28"
    ],
    "keywords": [
      "continuous measure",
      "turing degrees",
      "study degree-theoretic properties",
      "continuous probability measure",
      "generalized hausdorff measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}