{
  "id": "1507.03369",
  "version": "v1",
  "published": "2015-07-13T09:42:50.000Z",
  "updated": "2015-07-13T09:42:50.000Z",
  "title": "Realization of aperiodic subshifts and densities in groups",
  "authors": [
    "Nathalie Aubrun",
    "Sebastián Barbieri",
    "Stéphan Thomassé"
  ],
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "A Theorem of Gao, Jackson and Seward, originally conjectured to be false by Glasner and Uspenskij, asserts that every countable group admits a 2-coloring. A direct consequence of this result is that every countable group has a strongly aperiodic subshift on the alphabet $\\{0,1\\}$. In this article, we use Lov\\'asz local lemma to first give a new simple proof of this result, and second to prove the existence of a $G$-effective strongly aperiodic subshift for any finitely generated group $G$. We also study the problem of realizing densities in groups as a way of generalizing Sturmian sequences. This problem surprisingly turned out to be harder. We provide subshifts realizing any density only in the case of finitely generated amenable groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-13T09:42:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "realization",
      "lovasz local lemma",
      "direct consequence",
      "simple proof",
      "effective strongly aperiodic subshift"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}