{
  "abstract": "Let $p$ be an odd prime and let $A,B\\subseteq\\mathbb{F}_p$ be nonempty and distinct, with $|A|\\ge |B|$. We give a complete classification of the pairs for which the mixed restricted sumset $A\\mathbin{\\widehat+} B=\\{x+y:x\\in A,\\ y\\in B,\\ x\\ne y\\}$ has size $\\min\\{p,|A|+|B|-2\\}$. The main new contribution concerns the interior range $|A|+|B|\\le p-1$. For $|B|\\ge3$ and $|A|-|B|\\ge3$, criticality forces $A$ and $B$ to be common-endpoint arithmetic progressions, apart from a single common affine orbit with $(p,|A|,|B|)=(13,7,4)$. For $|A|-|B|\\in\\{1,2\\}$, criticality itself forces $B\\subsetneq A$, after which the restricted self-sum determines the deletion patterns. Together with the remaining cases, this yields the complete classification.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23343",
  "author_total": 2,
  "authors": [
    {
      "name": "Weilin Zhang"
    },
    {
      "name": "Hongjian Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7w9sw5wk0x3k5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23343",
  "fulltext_url": null,
  "id": "7w9sw5wk0x3k5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23343",
  "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|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214045224",
  "pdf_url": "https://arxiv.org/pdf/2609.23343",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7w9sw5wk0x3k5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23343",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23343"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Critical Pairs for Mixed Restricted Sumsets in Prime Fields",
  "title_text": "Critical Pairs for Mixed Restricted Sumsets in Prime Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/7w9sw5wk0x3k5",
  "venue": "arXiv",
  "versions": []
}