{
  "id": "2303.09878",
  "version": "v1",
  "published": "2023-03-17T10:53:03.000Z",
  "updated": "2023-03-17T10:53:03.000Z",
  "title": "Unique representations of integers by linear forms",
  "authors": [
    "Sándor Z. Kiss",
    "Csaba Sándor"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $k\\ge 2$ be an integer and let $A$ be a set of nonnegative integers. For a $k$-tuple of positive integers $\\underline{\\lambda} = (\\lambda_{1}, \\dots{} ,\\lambda_{k})$ with $1 \\le \\lambda_{1} < \\lambda_{2} < \\dots{} < \\lambda_{k}$, we define the additive representation function $R_{A,\\underline{\\lambda}}(n) = |\\{(a_{1}, \\dots{} ,a_{k})\\in A^{k}: \\lambda_{1}a_{1} + \\dots{} + \\lambda_{k}a_{k} = n\\}|$. For $k = 2$, Moser constructed a set $A$ of nonnegative integers such that $R_{A,\\underline{\\lambda}}(n) = 1$ holds for every nonnegative integer $n$. In this paper we characterize all the $k$-tuples $\\underline{\\lambda}$ and the sets $A$ of nonnegative integers with $R_{A,\\underline{\\lambda}}(n) = 1$ for every integer $n\\ge 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-17T10:53:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear forms",
      "unique representations",
      "nonnegative integer",
      "additive representation function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}