{
  "id": "1505.01194",
  "version": "v1",
  "published": "2015-05-05T21:51:34.000Z",
  "updated": "2015-05-05T21:51:34.000Z",
  "title": "On the number of weighted subsequences with zero-sum in a finite abelian group",
  "authors": [
    "Abílio Lemos",
    "Allan de Oliveira Moura"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1101.4492, arXiv:1308.3316 by other authors",
  "categories": [
    "math.NT"
  ],
  "abstract": "Suppose $G$ is a finite abelian group and $S=g_{1}\\cdots g_{l}$ is a sequence of elements in $G$. For any element $g$ of $G$ and $A\\subseteq\\mathbb{Z}\\backslash\\left\\{ 0\\right\\} $, let $N_{A,g}(S)$ denote the number of subsequences $T=\\prod_{i\\in I}g_{i}$ of $S$ such that $\\sum_{i\\in I}a_{i}g_{i}=g$ , where $I\\subseteq\\left\\{ 1,\\ldots,l\\right\\}$ and $a_{i}\\in A$. The purpose of this paper is to investigate the lower bound for $N_{A,0}(S)$. In particular, we prove that $N_{A,0}(S)\\geq2^{|S|-D_{A}(G)+1}$, where $D_{A}(G)$ is the smallest positive integer $l$ such that every sequence over $G$ of length at least $l$ has a nonempty $A$-zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-05T21:51:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite abelian group",
      "weighted subsequences",
      "lower bound",
      "smallest positive integer",
      "zero-sum subsequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}