{
  "id": "1708.05919",
  "version": "v1",
  "published": "2017-08-20T01:41:17.000Z",
  "updated": "2017-08-20T01:41:17.000Z",
  "title": "Rigidity, graphs and Hausdorff dimension",
  "authors": [
    "N. Chatzikonstantinou",
    "A. Iosevich",
    "S. Mkrtchyan",
    "J. Pakianathan"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "For a compact set $E \\subset \\mathbb R^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k+1$ points in $E$ such that the distance between a pair of points is specified if the corresponding vertices of $G$ are connected by an edge. We regard two such frameworks as equivalent if the specified distances are the same. We show that in a suitable sense the set of equivalences of such frameworks naturally embeds in ${\\mathbb R}^m$ where $m$ is the number of \"essential\" edges of $G$. We prove that there exists a threshold $s_k<d$ such that if the Hausdorff dimension of $E$ is greater than $s_k$, then the $m$-dimensional Hausdorff measure of the set of equivalences of $G$-frameworks is positive. The proof relies on combinatorial, topological and analytic considerations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-20T01:41:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B10"
    ],
    "keywords": [
      "hausdorff dimension",
      "dimensional hausdorff measure",
      "compact set",
      "equivalences",
      "analytic considerations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}