{
  "abstract": "We introduce an explicit deterministic collection of $N$ spherical points, $\\cP_N\\subset\\mathbb S^2$, that we call the {\\em deterministic Diamond points}. For every $0<α<2$ we prove that there exists a constant $C_α>0$ such that \\[ 0\\leq \\frac{2^{α+1}}{α+2}N^2-\\sum_{x,y\\in\\cP_N}|x-y|^α\\le C_αN^{1-α/2}. \\] This result shows that $\\cP_N\\subset\\mathbb S^2$ has a Riesz energy deficit of optimal order in this range of the parameter. In particular, for $α=1$, Stolarsky's invariance principle implies that $\\cP_N\\subset\\mathbb S^2$ has $L^2$ spherical cap discrepancy of asymptotically optimal order \\[ D_{L^2}^C(\\mathcal P_N)\\asymp N^{-3/4}, \\] making it, to our knowledge, the first explicit configuration proved to have this property. More generally, our result implies that the point set $\\cP_N$ has Sobolev $H^{s}(\\Sph)$ worst-case error of optimal order for all $1<s<2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27708",
  "author_total": 3,
  "authors": [
    {
      "name": "Carlos Beltrán"
    },
    {
      "name": "Jordi Marzo"
    },
    {
      "name": "Joaquim Ortega-Cerdà"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5pcbvgf7cewnt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27708",
  "fulltext_url": null,
  "id": "5pcbvgf7cewnt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27708",
  "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.CA"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7214148599",
  "pdf_url": "https://arxiv.org/pdf/2609.27708",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5pcbvgf7cewnt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27708",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27708"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A set of points with optimal $L^2$ spherical cap discrepancy",
  "title_text": "A set of points with optimal L^2 spherical cap discrepancy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/5pcbvgf7cewnt",
  "venue": "arXiv",
  "versions": []
}