{
  "abstract": "For a subset $A$ of a finite abelian group $G$, let $Σ_k(A)$ denote the set of sums of $k$ distinct elements of $A$, and let $Σ(A)=\\bigcup_{k\\ge1}Σ_k(A)$. The problem of determining when $Σ(A)=G$ goes back to Erdős and Heilbronn. Nguyen and Vu later formulated the corresponding fixed-length question: for a prescribed $k$, when does $Σ_k(A)=G$? For groups of odd order, we prove the sharp large-order threshold $|A|>|G|/3+1$, which forces $Σ_k(A)=G$ throughout the entire middle range $4\\le k\\le |A|-4$. This determines the limiting thresholds posed by Girard, Griffiths and Hamidoune. If $p$ is the least prime divisor of $|G|$, then, for fixed odd $p$ and sufficiently large $|G|$, the exact threshold $|A|\\ge |G|/p+p-1$ forces covering throughout the middle range beginning at $2p-2$. Thus, throughout this range, prescribing the number of summands does not change the sharp covering threshold. For $3\\le k\\le2p-3$, the restricted and ordinary $k$-fold covering problems have the same asymptotic threshold. As an application to MDS elliptic codes on an elliptic curve $E/\\F_q$, we sharpen the half-scale conjecture of Li, Wan and Zhang, obtaining the sharp $1/3$-scale bound when $|E(\\F_q)|$ is odd and, more generally, the exact bound determined by the least prime divisor of $|E(\\F_q)|$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14496",
  "author_total": 2,
  "authors": [
    {
      "name": "Dongchun Han"
    },
    {
      "name": "Yuan Ren"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2qtxx8gtr7xey.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14496",
  "fulltext_url": null,
  "id": "2qtxx8gtr7xey",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14496",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7221816176",
  "pdf_url": "https://arxiv.org/pdf/2610.14496",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2qtxx8gtr7xey.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14496",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14496"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "When Do Restricted Sumsets Cover a Finite Abelian Group?",
  "title_text": "When Do Restricted Sumsets Cover a Finite Abelian Group?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/2qtxx8gtr7xey",
  "venue": "arXiv",
  "versions": []
}