{
  "abstract": "Let $\\mathsf{d}$ be the asymptotic density on the positive integers $\\mathbb{N}^+$. We provide sufficient conditions on an infinite set $B\\subseteq \\mathbb{N}^+$ so that for each $α\\in [0,1]$ there exists a set $A\\subseteq \\mathbb{N}^+$ such that $\\mathsf{d}(A)=0$ and $\\mathsf{d}(A+B)=α$. For instance, if $(b_n: n \\in \\mathbb{N}^+)$ is the increasing enumeration of $B$, it is sufficient that $\\liminf_n b_{n+1}/b_n>1$. Moreover, we show that, if $|B\\cap [1,n]|\\ll \\log(n)^{1-\\varepsilon}$ for some $\\varepsilon>0$, then for each $α\\in [0,1]$ there exists a set $A\\subseteq \\mathbb{N}^+$ such that $\\mathsf{d}(A)=\\mathsf{d}(A+B)=α$. Conversely, we show that, for each $\\varepsilon>0$, there exists an infinite set $B\\subseteq \\mathbb{N}^+$ such that $|B\\cap [1,n]| \\ll n^{2/3+\\varepsilon}$ and $\\mathsf{d}(A+B) \\in \\{0,1\\}$ for every $A\\subseteq \\mathbb{N}^+$ such that $A+B$ admits asymptotic density.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29660",
  "author_total": 1,
  "authors": [
    {
      "name": "Paolo Leonetti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2zarf12v5fbqw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29660",
  "fulltext_url": null,
  "id": "2zarf12v5fbqw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29660",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214281863",
  "pdf_url": "https://arxiv.org/pdf/2609.29660",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2zarf12v5fbqw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29660",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29660"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the density of $A+B$ when $B$ has few elements",
  "title_text": "On the density of A+B when B has few elements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/2zarf12v5fbqw",
  "venue": "arXiv",
  "versions": []
}