{
  "abstract": "We study randomized constructions of quasi-uniform designs on a compact metric-measure space satisfying two-sided polynomial ball-growth conditions. Given integers $N\\le M$, we first draw $M$ independent candidate points and then retain the first $N$ points of a farthest-point traversal of the candidate set. We prove probabilistic non-asymptotic bounds on the mesh ratio and show that $M=Θ(N\\log N)$ is the sharp order of the candidate-pool size required for bounded mesh ratio: a sufficiently large multiple of $N\\log(N/δ)$ suffices with probability at least $1-δ$, whereas, if $M=o(N\\log N)$, the mesh ratio diverges in probability for every procedure that selects $N$ points from the same independent candidate pool. If $M/(N\\log M)\\to\\infty$, the upper bound on the mesh ratio for the exact farthest-point sampling (FPS) tends to $2$. We also quantify the effect of approximate FPS and analyze direct and fast implementations in spaces of bounded doubling dimension. Finally, using a single infinite stream of candidate points and increasing candidate budgets, we construct an almost surely quasi-uniform nested sequence; under supercritical oversampling and exact FPS, or more generally when the approximation factors tend to $1$, its mesh-ratio limit superior equals $2$ for compact positive-volume subsets of Euclidean space.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05986",
  "author_total": 2,
  "authors": [
    {
      "name": "Takashi Goda"
    },
    {
      "name": "Hengjun Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6y224618zkvyb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05986",
  "fulltext_url": null,
  "id": "6y224618zkvyb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05986",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7212064588",
  "pdf_url": "https://arxiv.org/pdf/2609.05986",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6y224618zkvyb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05986",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05986"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Quasi-uniform designs from random candidates: optimal candidate complexity and farthest-point sampling",
  "title_text": "Quasi-uniform designs from random candidates: optimal candidate complexity and farthest-point sampling",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/6y224618zkvyb",
  "venue": "arXiv",
  "versions": []
}