{
  "abstract": "We employ the set-theoretic methods yielding the consistency of the existence of nontrivial pairwise disjoint covers of the unit interval (or equivalently $\\mathbb{R}^n$ for $n\\in\\mathbb{N}\\setminus\\{0\\}$) by less than $2^ω$ closed sets in the context of covers of infinite dimensional Banach spaces $\\ell_1(κ)$ for infinite $κ$ under geometric conditions arising in tiling theory and in approximation theory. Specifically, for $κ=ω_1, ω_2$, using side-by-side Sacks forcing we prove the consistency of an arbitrarily large continuum above $κ$ with the existence of a highly disconnected proximinal set in $\\ell_1(κ)$, built from norm-compact pieces separated by a common positive distance, and the existence of a normal disjoint tiling of $\\ell_1(κ)$ by star-shaped bodies of the form $X+B$, where $X$ is norm compact and $B$ is the unit ball. We also prove that if $κ$ is an uncountable cardinal of countable cofinality, then $\\ell_1(κ)$ does not admit any normal disjoint tiling by bodies of the form $X+B$, where $X$ is closed and norm-separable and $B$ is the unit ball. These results complement a result of Klee of 1981 obtained for $κ$ satisfying $κ^ω=κ$, recent results of De Bernardi, Russo, Sezgek and Somaglia, and a $\\mathsf{ZFC}$ a machine-discovered result (included in the appendix) that there are no nontrivial discrete Chebyshev sets in $\\ell_1(κ)$ when $κ<2^ω$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35180",
  "author_total": 2,
  "authors": [
    {
      "name": "Piotr Koszmider"
    },
    {
      "name": "Piotr Szewczak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/269a4e82kdg92.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35180",
  "fulltext_url": null,
  "id": "269a4e82kdg92",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35180",
  "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.GN"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7214758660",
  "pdf_url": "https://arxiv.org/pdf/2609.35180",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/269a4e82kdg92.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35180",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35180"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On disjoint tilings of $\\ell_1(κ)$ by star-shaped bodies",
  "title_text": "On disjoint tilings of ℓ_1(κ) by star-shaped bodies",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/269a4e82kdg92",
  "venue": "arXiv",
  "versions": []
}