{
  "abstract": "Let $A\\subseteq[0, L]$ be a set of $k\\geqslant 9$ integers such that $0, L\\in A$ and $\\gcd(A)=1$. Define the $h$-fold restricted sumset $$h^{\\wedge}A=\\{a_1+\\cdots+a_h: a_i\\in A,\\ 1\\leqslant i\\leqslant h, \\ \\ a_i\\neq a_j\\ {\\rm for}\\ i\\neq j\\}. $$ In this paper, we prove that, for every $1\\leqslant h\\leqslant k-1$, we have $$|h^{\\wedge}A|\\geqslant h(k-h)-k+2+\\min\\{L, h(k-h)-\\mathbf{1}_h\\}, $$ where $\\mathbf{1}_h=1$ for $h\\in\\{2, k-2\\}$ and $\\mathbf{1}_h=0$ otherwise. The bound is sharp for every $1\\leqslant h\\leqslant k-1$ and the cardinality threshold $k\\geqslant 9$ cannot be decreased. We also give a lower bound for the union of two consecutive restricted sumsets in a cyclic group and characterize equality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25073",
  "author_total": 2,
  "authors": [
    {
      "name": "Yujie"
    },
    {
      "name": "Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/exyrez1qg9673.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25073",
  "fulltext_url": null,
  "id": "exyrez1qg9673",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25073",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223033367",
  "pdf_url": "https://arxiv.org/pdf/2610.25073",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/exyrez1qg9673.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25073",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25073"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A sharp lower bound for $h$-fold restricted sumsets of integers",
  "title_text": "A sharp lower bound for h-fold restricted sumsets of integers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/exyrez1qg9673",
  "venue": "arXiv",
  "versions": []
}