{
  "id": "1911.11963",
  "version": "v1",
  "published": "2019-11-27T05:37:17.000Z",
  "updated": "2019-11-27T05:37:17.000Z",
  "title": "Group topologies on integers and S-unit equations",
  "authors": [
    "Saveliy Skresanov"
  ],
  "categories": [
    "math.GR",
    "math.GN",
    "math.NT"
  ],
  "abstract": "A sequence of integers $ \\{ s_n \\}_{n \\in \\mathbb{N}} $ is called a T-sequence if there exists a Hausdorff group topology on $ \\mathbb{Z} $ such that $ \\{ s_n \\}_{n \\in \\mathbb{N}} $ converges to zero. For every finite set of primes $ S $ we build a Hausdorff group topology on $ \\mathbb{Z} $ such that every growing sequence of $ S $-integers converges to zero. As a corollary, we solve in the affirmative an open problem by I.V. Protasov and E.G. Zelenuk asking if $ \\{ 2^n + 3^n \\}_{n \\in \\mathbb{N}} $ is a T-sequence. Our results rely on a nontrivial number-theoretic fact about $ S $-unit equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-27T05:37:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54H11",
      "11D61",
      "11Z05"
    ],
    "keywords": [
      "s-unit equations",
      "hausdorff group topology",
      "nontrivial number-theoretic fact",
      "t-sequence",
      "finite set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}