{
  "id": "1607.02849",
  "version": "v1",
  "published": "2016-07-11T08:04:36.000Z",
  "updated": "2016-07-11T08:04:36.000Z",
  "title": "Affine embeddings of Cantor sets on the line",
  "authors": [
    "Amir Algom"
  ],
  "comment": "10 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $s\\in (0,1)$, and let $F\\subset \\mathbb{R}$ be a self similar set such that $0 < \\dim_H F \\leq s$ . We prove that there exists $\\delta= \\delta(s) >0$ such that if $F$ admits an affine embedding into a homogeneous self similar set $E$ and $0 \\leq \\dim_H E - \\dim_H F < \\delta$ then (under some mild conditions on $E$ and $F$) the contraction ratios of $E$ and $F$ are logarithmically commensurable. This provides more evidence for a Conjecture of Feng, Huang, and Rao, that states that these contraction ratios are logarithmically commensurable whenever $F$ admits an affine embedding into $E$ (under some mild conditions). Our method is a combination of an argument based on the approach of Feng, Huang, and Rao, with a new result by Hochman, which is related to the increase of entropy of measures under convolutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-11T08:04:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "affine embedding",
      "cantor sets",
      "mild conditions",
      "contraction ratios",
      "homogeneous self similar set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}