From Erdos Problem 1154 to a Zero One Law for Turing Ideals

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.