{
  "id": "2203.02665",
  "version": "v1",
  "published": "2022-03-05T06:22:42.000Z",
  "updated": "2022-03-05T06:22:42.000Z",
  "title": "On unit weighted zero-sum constants of $\\mathbb Z_n$",
  "authors": [
    "Santanu Mondal",
    "Krishnendu Paul",
    "Shameek Paul"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The $A$-weighted Gao constant $E_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\\mathbb Z_n$ has a subsequence of length $n$, whose $A$-weighted sum is zero. When $A=U(n)$ (the set of all units in $\\mathbb Z_n$), the value of $E_A(n)$ has been determined by Simon Griffiths and by Florian Luca. We give another proof of this result and also determine the values of two related constants $C_A(n)$ and $D_A(n)$, by using the corresponding constants $E(\\mathbb Z_2^a),C(\\mathbb Z_2^a)$ and $D(\\mathbb Z_2^a)$ for the group $\\mathbb Z_2^a$. We also characterize all sequences of length $E_A(n)-1$ in $\\mathbb Z_n$, which do not have any $A$-weighted zero-sum subsequence of length $n$, when $n$ is a power of 2 and $A=U(n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-05T06:22:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B50",
      "11B75"
    ],
    "keywords": [
      "unit weighted zero-sum constants",
      "smallest natural number",
      "weighted gao constant",
      "weighted zero-sum subsequence",
      "florian luca"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}