{
  "abstract": "Let $m(p)$ be the least size of a subset of $\\F_p$ with at least two elements for which every sum has two distinct representations as unordered pairs, allowing repetition. We prove that, for every prime $p\\ge64$, \\[ m(p)\\ge2^{-80}\\left(\\frac{\\log p}{\\log\\log p}\\right)^2. \\] The argument compresses the full integer collision lattice by unit-pivot elimination. A shared random sample and forests of bounded diameter give $O(\\sqrt n+n/\\log p)$ surviving coordinates of polynomial height for a minimal set of size $n$. A nonzero minor divisible by $p$ then gives the lower bound. We also construct weakly ternary-balanced seeds yielding \\[ m(p)\\le\\frac{(\\log p)^2}{2(\\log3)^2} +\\left(\\frac1{4\\log3}+o(1)\\right) \\frac{(\\log p)^2}{\\log\\log p}. \\] Consequently $m(p)=(\\log p)^{2+o(1)}$ as $p$ tends to infinity through the primes. The constant-factor order of $m(p)$ remains undetermined.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09349",
  "author_total": 2,
  "authors": [
    {
      "name": "Jianfeng Hou"
    },
    {
      "name": "Kai Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/420mjev1rf8b9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09349",
  "fulltext_url": null,
  "id": "420mjev1rf8b9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09349",
  "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|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220921284",
  "pdf_url": "https://arxiv.org/pdf/2610.09349",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/420mjev1rf8b9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09349",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09349"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A near-quadratic lower bound for sets with no unique sums",
  "title_text": "A near-quadratic lower bound for sets with no unique sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/420mjev1rf8b9",
  "venue": "arXiv",
  "versions": []
}