{
  "abstract": "Assume $V=L$. We prove two uniform theorems producing coanalytic sets of prescribed Hausdorff dimension in Euclidean space. The first concerns countable families of partial operations of finite arity on $\\mathbb{R}^n$ whose evaluation is hyperarithmetic and which are tame in each vector variable separately. For such families a single lightface $Π^1_1$ relation yields strictly increasing, left-continuous chains of proper sets closed under the operations, one of each Hausdorff dimension in $[0,n]$, uniformly in the family and in $n$; no exchange property of the closure is needed. As an application, there are coanalytic proper subfields of $\\mathbb{R}$ of every Hausdorff dimension in $[0,1]$. The second theorem constructs coanalytic bases of every possible Hausdorff dimension for effectively coded pregeometries. As an application, there are coanalytic Hamel bases of $\\mathbb{R}^n$ and transcendence bases of $\\mathbb{C}$ of every Hausdorff dimension in $[0,n]$ and $[0,2]$, respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09707",
  "author_total": 1,
  "authors": [
    {
      "name": "Yi Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/axk8bqdq1pbpp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09707",
  "fulltext_url": null,
  "id": "axk8bqdq1pbpp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09707",
  "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.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7221003017",
  "pdf_url": "https://arxiv.org/pdf/2610.09707",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/axk8bqdq1pbpp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.09707",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09707"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Coanalytic subfields and bases of prescribed dimension",
  "title_text": "Coanalytic subfields and bases of prescribed dimension",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/axk8bqdq1pbpp",
  "venue": "arXiv",
  "versions": []
}