Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Wang, Yi
Institution: Tsinghua University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221003017