{
  "abstract": "Let $B^d$ denote the $d$-dimensional Euclidean ball of unit radius. What is the largest constant $f(B^d) \\in [0,1]$ with the property that every finite collection $\\mathcal{C}$ of unit balls in $\\mathbb{R}^d$ admits a disjoint sub-collection $\\mathcal{S}$ occupying at least a fraction $f(B^d)$ of the volume of $\\mathcal{C}$? This problem was first raised by T. Radó in 1928, for axis-parallel squares in the plane; the author was motivated by a classical covering lemma in real analysis due to Vitali. The case of Euclidean balls was first considered by R. Rado in 1949. Until last year the best known estimates on $f(B^d)$ for unit balls where very far apart: \\[ (1+ε_d) 3^{-d} \\leq f(B^d) \\leq 2^{-d}, \\] where $0<ε_d=o_{d\\rightarrow \\infty}(1)$. Recently, the authors of this note observed that an exponential improvement on the upper bound follows from the Kabatiansky--Levenshtein spherical code bound, while the lower bound was improved by a linear factor by C.~Xie and G.~Ge (see arxiv:2608.09744). The current best estimates for large $d$ are \\[ c \\cdot d \\cdot 3^{-d} \\leq f(B^d) \\leq 2.447^{-d}, \\] where $c>0$ is an absolute constant. Here we offer the first exponential improvement of the lower bound in almost 80 years, which narrows the gap to: \\[ 2.910^{-d} \\leq f(B^d) \\leq 2.447^{-d}. \\] Our method is constructive and yields a polynomial time algorithm for finding a disjoint sub-collection realizing the estimate. Moreover the same technique gives similar exponentially improved lower bounds for all symmetric convex bodies satisfying a uniform convexity assumption, e.g., $\\ell^p$-balls for all $p\\in (1,\\infty)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25288",
  "author_total": 2,
  "authors": [
    {
      "name": "Gian Maria Dall'Ara"
    },
    {
      "name": "Adrian Dumitrescu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aqnwgc2tqj5pf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25288",
  "fulltext_url": null,
  "id": "aqnwgc2tqj5pf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25288",
  "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",
      "math.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7214103878",
  "pdf_url": "https://arxiv.org/pdf/2609.25288",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aqnwgc2tqj5pf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.25288",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25288"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Exponential improvements in Rado's covering problem",
  "title_text": "Exponential improvements in Rado's covering problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/aqnwgc2tqj5pf",
  "venue": "arXiv",
  "versions": []
}