{
  "id": "2311.00090",
  "version": "v1",
  "published": "2023-10-31T18:58:31.000Z",
  "updated": "2023-10-31T18:58:31.000Z",
  "title": "Smooth weighted zero-sum constants",
  "authors": [
    "Krishnendu Paul",
    "Shameek Paul"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $A\\subseteq\\mathbb Z_n$ be a weight-set and $S=(x_1,x_2,\\ldots, x_k)$ be a sequence in $\\mathbb Z_n$. We say that $S$ is a smooth $A$-weighted zero-sum sequence if there exists $(a_1,\\ldots,a_k)\\in A^k$ such that we have $a_1x_1+\\cdots+a_kx_k=0$ and $a_1+\\cdots+a_k=0$. It is easy to see that if $S$ is a smooth $A$-weighted zero-sum sequence, then for every $y\\in \\mathbb Z_n$ the sequence $S+y=(x_1+y,\\ldots,x_k+y)$ is also a smooth $A$-weighted zero-sum sequence. From the well known EGZ-theorem it follows that if $S$ has length at least $2n-1$, then $S$ has a smooth $A$-weighted zero-sum subsequence of length $n$. The constant $\\bar E_A$ is defined to be the smallest positive integer $k$ such that any sequence of length $k$ in $\\mathbb Z_n$ has a smooth $A$-weighted zero-sum subsequence of length $n$. A sequence in $\\mathbb Z_n$ of length $\\bar E_A-1$ which does not have any smooth $A$-weighted zero-sum subsequence of length $n$ is called an $\\bar E$-extremal sequence for $A$. For every $n$ we consider the weight-sets $\\{1\\}$ and $\\mathbb Z_n\\setminus\\{0\\}$. When $n$ is an odd prime $p$ we consider the weight-set $Q_p$ of all non-zero quadratic residues. We also study the related constants $\\bar C_A$ and $\\bar D_A$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-31T18:58:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B50"
    ],
    "keywords": [
      "smooth weighted zero-sum constants",
      "weighted zero-sum sequence",
      "weighted zero-sum subsequence",
      "weight-set",
      "non-zero quadratic residues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}