{
  "abstract": "For a set $A$ of positive integers, let $d_A(n)$ denote the number of representations of $n$ as a difference of two elements of $A$. We prove that, if $\\liminf_{n\\to\\infty}f(n)\\geq2$, then for every Sidon set $B$ and every positive function $ω(x)$ tending to infinity there is a set $A\\subseteq\\N$ such that $d_A(n)=f(n)$ for all $n\\in\\N$ and $|A(x)-B(x/2)|\\leqω(x)$ for all $x\\geq1$. We also give two extensions that allow $f(n)=1$ for infinitely many $n$. For an infinite Sidon set $B$, the same density scale $B(x/2)$ is retained when $f$ is at least $2$ on arbitrarily long intervals centered at elements of $2B$. If $\\{n:f(n)\\geq2\\}$ contains arbitrarily long intervals, the density scale $B(x/3)$ is obtained for every Sidon set $B$. The proofs use remote insertion of new representations together with controlled deletions from a Sidon core. As a quantitative consequence, Ruzsa's dense infinite Sidon sequence gives $A(x)=x^{\\sqrt2-1+o(1)}$ whenever $\\{n:f(n)\\geq2\\}$ contains arbitrarily long intervals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27786",
  "author_total": 2,
  "authors": [
    {
      "name": "Wu-Xia Ma"
    },
    {
      "name": "Bing-Ling Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40dvpz57jpe5r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27786",
  "fulltext_url": null,
  "id": "40dvpz57jpe5r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27786",
  "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": "W7223088044",
  "pdf_url": "https://arxiv.org/pdf/2610.27786",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40dvpz57jpe5r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27786",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27786"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Dense Sets with Prescribed Difference Representation Functions",
  "title_text": "Dense Sets with Prescribed Difference Representation Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/40dvpz57jpe5r",
  "venue": "arXiv",
  "versions": []
}