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Coanalytic subfields and bases of prescribed dimension

  • Yi Wang
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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 ℝn whose evaluation is hyperarithmetic and which are tame in each vector variable separately. For such families a single lightface Π11 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 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 ℝn and transcendence bases of ℂ of every Hausdorff dimension in [0,n] and [0,2], respectively.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026coanalytic,
  title = {{Coanalytic subfields and bases of prescribed dimension}},
  author = {Yi Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09707},
  eprint = {2610.09707},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09707},
}
RIS
TY  - UNPB
TI  - Coanalytic subfields and bases of prescribed dimension
AU  - Yi Wang
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.09707
UR  - https://arxiv.org/abs/2610.09707
AB  - 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 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 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 ℝ^n and transcendence bases of ℂ of every Hausdorff dimension in [0,n] and [0,2], respectively.
ER  -