{
  "id": "2209.10823",
  "version": "v1",
  "published": "2022-09-22T07:11:14.000Z",
  "updated": "2022-09-22T07:11:14.000Z",
  "title": "Sets of full measure avoiding Cantor sets",
  "authors": [
    "Mihail N. Kolountzakis"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "In relation to the Erd\\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are not universal in measure, i.e. they satisfy the above conjecture. These are symmetric Cantor sets $C$ which can be quite thin: the length of the $n$-th generation intervals defining the Cantor set is decreasing almost doubly exponentially. Further, we achieve to construct a set, not just of positive measure, but of \\textit{full measure} not containing any affine copy of $C$. Our method is probabilistic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-22T07:11:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "05D40"
    ],
    "keywords": [
      "full measure avoiding cantor sets",
      "affine copy",
      "infinite set",
      "os similarity problem",
      "symmetric cantor sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}