AI 中文总结
该研究针对埃尔德什问题1154，结合数字交错与Orponen和Shmerkin的Furstenberg集定理，证明了任意图灵理想的豪斯多夫维数为0或1，为内模型实数的维数问题提供了新结论。
AI 中文摘要
埃尔德什问题1154询问[0,1]中的每个数是否都可作为实数的子环或子域的豪斯多夫维数。受该问题启发，喻良提出：当在内模型的实数在外模型中计算维数时，其内模型的实数的豪斯多夫维数是否能严格介于0和1之间。我们证明了一个更强的结果：若ℐ⊆2^ω是任意一个图灵理想，则dim_H ℐ∈{0,1}。等价地，其实数具有图灵度属于ℐ的实闭域的豪斯多夫维数为0或1。该证明将数字交错与Orponen和Shmerkin的Furstenberg集定理相结合。
英文摘要
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 $\mathcal I \subseteq 2^ω$ 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.
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