{
  "id": "math/0602568",
  "version": "v1",
  "published": "2006-02-25T05:39:46.000Z",
  "updated": "2006-02-25T05:39:46.000Z",
  "title": "Long zero-free sequences in finite cyclic groups",
  "authors": [
    "Svetoslav Savchev",
    "Fang Chen"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than $n/2$ in the additive group $\\Zn/$ of integers modulo $n$. The main result states that for each zero-free sequence $(a_i)_{i=1}^\\ell$ of length $\\ell>n/2$ in $\\Zn/$ there is an integer $g$ coprime to $n$ such that if $\\bar{ga_i}$ denotes the least positive integer in the congruence class $ga_i$ (modulo $n$), then $\\Sigma_{i=1}^\\ell\\bar{ga_i}<n$. The answers to a number of frequently asked zero-sum questions for cyclic groups follow as immediate consequences. Among other applications, best possible lower bounds are established for the maximum multiplicity of a term in a zero-free sequence with length greater than $n/2$, as well as for the maximum multiplicity of a generator. The approach is combinatorial and does not appeal to previously known nontrivial facts.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-02-25T05:39:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B50",
      "11P21"
    ],
    "keywords": [
      "finite cyclic groups",
      "long zero-free sequences",
      "maximum multiplicity",
      "main result states",
      "additively written abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......2568S"
    }
  }
}