{
  "id": "1307.6399",
  "version": "v5",
  "published": "2013-07-24T12:18:54.000Z",
  "updated": "2014-07-02T17:06:33.000Z",
  "title": "Self similar sets, entropy and additive combinatorics",
  "authors": [
    "Michael Hochman"
  ],
  "comment": "21 pages, 2 figures; submitted to Proceedings of AFRT 2012. v5: more typos corrected",
  "categories": [
    "math.CA",
    "math.DS"
  ],
  "abstract": "This article is an exposition of recent results on self-similar sets, asserting that if the dimension is smaller than the trivial upper bound then there are almost overlaps between cylinders. We give a heuristic derivation of the theorem using elementary arguments about covering numbers. We also give a short introduction to additive combinatorics, focusing on inverse theorems, which play a pivotal role in the proof. Our elementary approach avoids many of the technicalities in the original proof but also falls short of a complete proof. In the last section we discuss how the heuristic argument is turned into a rigorous one.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2014-07-02T17:06:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "11K55",
      "11B30",
      "11P70"
    ],
    "keywords": [
      "self similar sets",
      "additive combinatorics",
      "elementary approach avoids",
      "trivial upper bound",
      "original proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.6399H"
    }
  }
}