{
  "id": "1012.4177",
  "version": "v1",
  "published": "2010-12-19T16:00:25.000Z",
  "updated": "2010-12-19T16:00:25.000Z",
  "title": "A Kronecker-Weyl theorem for subsets of abelian groups",
  "authors": [
    "Dikran Dikranjan",
    "Dmitri Shakhmatov"
  ],
  "journal": "Advances in Mathematics, 226 (2011), 4776-4795",
  "doi": "10.1016/j.aim.2010.12.016",
  "categories": [
    "math.GR",
    "math.DS",
    "math.GN",
    "math.LO",
    "math.NT"
  ],
  "abstract": "Let N be the set of non-negative integer numbers, T the circle group and c the cardinality of the continuum. Given an abelian group G of size at most 2^c and a countable family F of infinite subsets of G, we construct \"Baire many\" monomorphisms p: G --> T^c such that p(E) is dense in {y in T^c : ny=0} whenever n in N, E in F, nE={0} and {x in E: mx=g} is finite for all g in G and m such that n=mk for some k in N--{1}. We apply this result to obtain an algebraic description of countable potentially dense subsets of abelian groups, thereby making a significant progress towards a solution of a problem of Markov going back to 1944. A particular case of our result yields a positive answer to a problem of Tkachenko and Yaschenko. Applications to group actions and discrete flows on T^c, diophantine approximation, Bohr topologies and Bohr compactifications are also provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-19T16:00:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20K30",
      "03E15",
      "11K36",
      "11K60",
      "22A05",
      "37B05",
      "54D65",
      "54E52"
    ],
    "keywords": [
      "abelian group",
      "kronecker-weyl theorem",
      "bohr compactifications",
      "non-negative integer numbers",
      "infinite subsets"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.4177D"
    }
  }
}