{
  "abstract": "Let $A$ and $B$ be infinite sets of nonnegative integers whose sum contains all sufficiently large integers, and let $A(x)$ and $B(x)$ be their counting functions. We give a discrete proof of the sharp implication \\[ \\limsup_{x\\to\\infty}\\frac{A(x)B(x)}{x}<\\frac32 \\quad\\Longrightarrow\\quad A(x)B(x)-x\\longrightarrow+\\infty. \\] The constant $3/2$ is best possible. Using finite convolution sums, we show that a long gap in one set forces a gap in the other. We then compare the counting functions before and after the gap to obtain the sharp bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29963",
  "author_total": 1,
  "authors": [
    {
      "name": "Zihan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6ek0kc6591p39.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29963",
  "fulltext_url": null,
  "id": "6ek0kc6591p39",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29963",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223198262",
  "pdf_url": "https://arxiv.org/pdf/2610.29963",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6ek0kc6591p39.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29963",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29963"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Sharp Bound for Additive Complements",
  "title_text": "A Sharp Bound for Additive Complements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/6ek0kc6591p39",
  "venue": "arXiv",
  "versions": []
}