{
  "abstract": "Let $\\mathsf{d}$, $\\mathsf{d}^{\\ast}$, and $\\mathsf{d}_{\\ast}$ denote, respectively, the asymptotic density, the upper asymptotic density, and the lower asymptotic density on $\\mathbb N$. We show that there are no constants $0<ν<1$ and $c>0$ such that $$ \\mathsf{d}_{\\ast}(A+A)\\ge c\\bigl(\\mathsf{d}^{\\ast}(A+A)\\bigr)^{1-ν}\\mathsf{d}(A)^ν $$ for every set $A\\subseteq \\mathbb{N}$ which admits asymptotic density. This answers in the negative a question of I.Z. Ruzsa.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20206",
  "author_total": 1,
  "authors": [
    {
      "name": "Paolo Leonetti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2j02eh6g7nahn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20206",
  "fulltext_url": null,
  "id": "2j02eh6g7nahn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20206",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213602219",
  "pdf_url": "https://arxiv.org/pdf/2609.20206",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2j02eh6g7nahn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20206",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20206"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On a question of Ruzsa about densities of sumsets",
  "title_text": "On a question of Ruzsa about densities of sumsets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/2j02eh6g7nahn",
  "venue": "arXiv",
  "versions": []
}