{
  "id": "1709.05266",
  "version": "v1",
  "published": "2017-09-15T15:23:57.000Z",
  "updated": "2017-09-15T15:23:57.000Z",
  "title": "Dimension 1 sequences are close to randoms",
  "authors": [
    "Noam Greenberg",
    "Joe Miller",
    "Alexander Shen",
    "Linda Brown Westrick"
  ],
  "comment": "19 pages",
  "categories": [
    "math.LO",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "We show that a sequence has effective Hausdorff dimension 1 if and only if it is coarsely similar to a Martin-L\\\"{o}f random sequence. More generally, a sequence has effective dimension $s$ if and only if it is coarsely similar to a weakly $s$-random sequence. Further, for any $s<t$, every sequence of effective dimension $s$ can be changed on density at most $H^{-1}(t)-H^{-1}(s)$ of its bits to produce a sequence of effective dimension $t$, and this bound is optimal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-15T15:23:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03D32",
      "68Q30"
    ],
    "keywords": [
      "effective dimension",
      "random sequence",
      "coarsely similar",
      "effective hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}