{
  "abstract": "Let $p$ be an odd prime, and write $A\\RS B=\\{a+b:a\\in A,\\ b\\in B,\\ a\\ne b\\}$. We classify all pairs $A,B\\subseteq\\Fp$ satisfying $|A|+|B|=p$, $|A|>|B|=\\ell\\ge1$, and $|A\\RS B|=p-2$. After normalizing the two missing sums to $\\{0,1\\}$, we obtain exactly $\\ell+1$ explicit models. The classification forces $B\\subseteq A$ and yields, for fixed $p,\\ell$, exactly $\\binom p2(\\ell+1)$ ordered pairs and $1+\\lfloor\\ell/2\\rfloor$ equivalence classes under simultaneous affine transformations. We determine when either set is an arithmetic progression and compute exact difference-set cardinalities. For $\\ell\\ge3$ and $p\\ge4\\ell-5$, the cardinality $|B-B|$ determines the equivalence class at fixed $p,\\ell$. The proof uses punctured translates and cyclic component counting. We also exhibit a critical pair in $\\mathbb F_{13}$, with size gap three and $|A|+|B|=p-2$, that contradicts a proposed inverse statement below the boundary.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14544",
  "author_total": 3,
  "authors": [
    {
      "name": "Hongjian Li"
    },
    {
      "name": "Yangcheng Li"
    },
    {
      "name": "Pingzhi Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7r9747h4ydvp0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14544",
  "fulltext_url": null,
  "id": "7r9747h4ydvp0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14544",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213298421",
  "pdf_url": "https://arxiv.org/pdf/2609.14544",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7r9747h4ydvp0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14544",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14544"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Critical Restricted Sumsets at the Boundary $\\lvert A\\rvert+\\lvert B\\rvert=p$",
  "title_text": "Critical Restricted Sumsets at the Boundary lvert Arvert+lvert Brvert=p",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/7r9747h4ydvp0",
  "venue": "arXiv",
  "versions": []
}