{
  "id": "0912.0497",
  "version": "v1",
  "published": "2009-12-02T19:05:35.000Z",
  "updated": "2009-12-02T19:05:35.000Z",
  "title": "Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density",
  "authors": [
    "Dikran Dikranjan",
    "Dmitri Shakhmatov"
  ],
  "journal": "Proceedings of the American Mathematical Society, 138 (2010), 2979-2990",
  "doi": "10.1090/S0002-9939-10-10302-5",
  "categories": [
    "math.GN",
    "math.GR"
  ],
  "abstract": "For an uncountable cardinal \\tau and a subset S of an abelian group G, the following conditions are equivalent: (i) |{ns:s\\in S}|\\ge \\tau for all integers n\\ge 1; (ii) there exists a group homomorphism \\pi:G\\to T^{2^\\tau} such that \\pi(S) is dense in T^{2^\\tau}. Moreover, if |G|\\le 2^{2^\\tau}, then the following item can be added to this list: (iii) there exists an isomorphism \\pi:G\\to G' between G and a subgroup G' of T^{2^\\tau} such that \\pi(S) is dense in T^{2^\\tau}. We prove that the following conditions are equivalent for an uncountable subset S of an abelian group G that is either (almost) torsion-free or divisible: (a) S is T-dense in G for some Hausdorff group topology T on G; (b) S is T-dense in some precompact Hausdorff group topology T on G; (c) |{ns:s\\in S}|\\ge \\min{\\tau:|G|\\le 2^{2^\\tau}} for every integer n\\ge 1. This partially resolves a question of Markov going back to 1946.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-02T19:05:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22A05",
      "20K99",
      "22C05",
      "54A25",
      "54B10",
      "54D65"
    ],
    "keywords": [
      "markovs potential density",
      "hewitt-marczewski-pondiczery type theorem",
      "abelian group",
      "precompact hausdorff group topology",
      "equivalent"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.0497D"
    }
  }
}