{
  "id": "1411.7116",
  "version": "v1",
  "published": "2014-11-26T06:30:07.000Z",
  "updated": "2014-11-26T06:30:07.000Z",
  "title": "Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets",
  "authors": [
    "Hui Rao",
    "Yuan Zhang"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The higher dimensional Frobenius problem was introduced by a preceding paper [Fan, Rao and Zhang, Higher dimensional Frobenius problem: maximal saturated cones, growth function and rigidity, Preprint 2014]. %the higher dimensional Frobenius problem was introduced and a directional growth function was studied. In this paper, we investigate the Lipschitz equivalence of dust-like self-similar sets in $\\mathbb R^d$. For any self-similar set, we associate with it a higher dimensional Frobenius problem, and we show that the directional growth function of the associate higher dimensional Frobenius problem is a Lipschitz invariant. As an application, we solve the Lipschitz equivalence problem when two dust-like self-similar sets $E$ and $F$ have coplanar ratios, by showing that they are Lipschitz equivalent if and only if the contraction vector of the $p$-th iteration of $E$ is a permutation of that of the $q$-th iteration of $F$ for some $p, q\\geq 1$. This partially answers a question raised by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, \\emph{Mathematika,} \\textbf{39} (1992), 223--233].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-26T06:30:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lipschitz equivalence",
      "cantor sets",
      "directional growth function",
      "dust-like self-similar sets",
      "associate higher dimensional frobenius problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.7116R"
    }
  }
}