{
  "abstract": "A Sidon set is a set $\\mathcal{A}$ of integers with the property that if $0\\neq a-b=c-d$ and $a,b,c,d\\in\\mathcal{A}$, then $(a,b)=(c,d)$. Our main result is an improved and explicit bound on the diameter of a Sidon set: if $\\mathcal{A}$ is a Sidon set with $k\\ge 1$ elements, then \\[\\operatorname{diam}(\\mathcal{A}) \\ge k^2 \\left(\\frac{1}{1 + 0.945/\\sqrt{k}}-\\frac{0.945}{\\sqrt k}\\right) -1;\\] in particular, $\\operatorname{diam}(\\mathcal{A})\\ge k^2-1.89 k^{3/2}$. This implies that if $\\mathcal{A}\\subseteq[1,N]$ is a Sidon set, then $|\\mathcal{A}| \\le N^{1/2}+0.945 \\, N^{1/4}+0.447$. Our technique also gives an improved bound for the case of large $k$, namely $\\operatorname{diam}(\\mathcal{A}) \\ge k^2-1.88607 k^{3/2}$ for $k\\ge 483$. A $γ$-Golomb ruler is a set $\\mathcal{A}$ of integers with the property that there are not $γ+1$ distinct pairs $(a_i,b_i)\\in\\mathcal{A}^2$ with $0\\neq a_1-b_1=\\cdots=a_{γ+1}-b_{γ+1}$. Our diameter bounds extend in a natural way to the case of Golomb rulers. Our techniques can also be adapted to address a number of questions on the distribution of Golomb rulers: we provide upper bounds on the two quantities $\\left|a_\\ell-a_1 -\\frac{\\ell-1}{k-1}(a_k-a_1)\\right|$ and $\\max_{1\\le i<k} (a_{i+1}-a_i)$, where $\\mathcal{A}=\\{a_1<\\dots<a_k\\}$ is a $γ$-Golomb ruler with $\\operatorname{diam}(\\mathcal{A})\\le (k^2-bk^{3/2})/γ$ (where $b\\in \\mathbb{R}$ is a free parameter).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36207",
  "author_total": 3,
  "authors": [
    {
      "name": "Daniel Carter"
    },
    {
      "name": "Kevin O'Bryant"
    },
    {
      "name": "Terence Tao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3sv1v36t6gys4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36207",
  "fulltext_url": null,
  "id": "3sv1v36t6gys4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36207",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7224118100",
  "pdf_url": "https://arxiv.org/pdf/2610.36207",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3sv1v36t6gys4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36207",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36207"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the diameter and distribution of Sidon sets",
  "title_text": "On the diameter and distribution of Sidon sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/3sv1v36t6gys4",
  "venue": "arXiv",
  "versions": []
}