[2608.18955] From Erdos Problem 1154 to a Zero One Law for Turing Ideals
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arXiv:2608.18955 (math)
[Submitted on 19 Aug 2026]
Title:From Erdos Problem 1154 to a Zero One Law for Turing Ideals
Authors:Yi Wang<br>View a PDF of the paper titled From Erdos Problem 1154 to a Zero One Law for Turing Ideals, by Yi Wang
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Abstract:Erdős Problem 1154 asks whether every number in $[0,1]$ occurs as the Hausdorff dimension of a subring or subfield of $\mathbb{R}$. Motivated by this problem, Liang Yu asked whether the reals of an inner model can have Hausdorff dimension strictly between zero and one when their dimension is computed in an outer model. We prove a stronger result: if $\mathcal I \subseteq 2^\omega$ is any Turing ideal, then $\dim_{\mathrm H} \mathcal I \in \{0,1\}$. Equivalently, the real-closed field whose reals have Turing degrees in $\mathcal I$ has Hausdorff dimension zero or one. The proof combines digit interleaving with the Furstenberg-set theorem of Orponen and Shmerkin.
Comments:<br>6 pages, no figures
Subjects:
Logic (math.LO); Classical Analysis and ODEs (math.CA)
MSC classes:<br>28A78
Cite as:<br>arXiv:2608.18955 [math.LO]
(or<br>arXiv:2608.18955v1 [math.LO] for this version)
https://doi.org/10.48550/arXiv.2608.18955
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arXiv-issued DOI via DataCite (pending registration)
Submission history<br>From: Yi Wang [view email]<br>[v1]<br>Wed, 19 Aug 2026 14:19:51 UTC (5 KB)
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