{
  "abstract": "The least interval diameter needed to realize every attainable cardinality of an $h$-fold sumset of a four-element integer set is $\\binom{h+2}{2}+(h\\bmod2)$ for $2\\le h\\le10$ and $d_h=(h-3)(h-1)$ for $h\\ge11$. We prove the formula uniformly for $h\\ge17$ and use complete finite searches, with a relation-rank coverage argument, for the remaining orders. For the lower bound, we show that every integer four-set with $\\binom{h+3}{3}-17$ distinct $h$-fold sums has normalized diameter in $\\{d_h,d_h+1,d_h+2\\}$ when $h\\ge9$, and we construct sets attaining $d_h$. For the upper bound, a relation-lattice index argument restricts larger minimum-diameter representatives to two or three active directions. Two general constructions and four special defects then provide short sets realizing every remaining cardinality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02216",
  "author_total": 1,
  "authors": [
    {
      "name": "Enkai Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5xm1r6gz4t70e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02216",
  "fulltext_url": null,
  "id": "5xm1r6gz4t70e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02216",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220453305",
  "pdf_url": "https://arxiv.org/pdf/2610.02216",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5xm1r6gz4t70e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02216",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02216"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact diameter compression for four-point sumset cardinalities",
  "title_text": "Exact diameter compression for four-point sumset cardinalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/5xm1r6gz4t70e",
  "venue": "arXiv",
  "versions": []
}