{
  "id": "2211.13675",
  "version": "v1",
  "published": "2022-11-24T15:44:52.000Z",
  "updated": "2022-11-24T15:44:52.000Z",
  "title": "On Perfect Bases in Finite Abelian Groups",
  "authors": [
    "Bela Bajnok",
    "Connor Berson",
    "Hoang Anh Just"
  ],
  "comment": "To appear in Involve",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $G$ be a finite abelian group and $s$ be a positive integer. A subset $A$ of $G$ is called a {\\em perfect $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ (not-necessarily-distinct) elements of $A$; similarly, we say that $A$ is a {\\em perfect restricted $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ distinct elements of $A$. We prove that perfect $s$-bases exist only in the trivial cases of $s=1$ or $|A|=1$. The situation is different with restricted addition where perfection is more frequent; here we treat the case of $s=2$ and prove that $G$ has a perfect restricted $2$-basis if, and only if, it is isomorphic to $\\mathbb{Z}_2$, $\\mathbb{Z}_4$, $\\mathbb{Z}_7$, $\\mathbb{Z}_2^2$, $\\mathbb{Z}_2^4$, or $\\mathbb{Z}_2^2 \\times \\mathbb{Z}_4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-24T15:44:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite abelian group",
      "perfect bases",
      "distinct elements",
      "trivial cases",
      "restricted addition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}