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R. Daniel Mauldin, Subfields of $\mathbf{R}$ with arbitrary Hausdorff dimension, Math. Proc. Cambridge Philos. Soc. 161 (2016), no. 1, 157-165, published online 2016-03-31. The paper proves, assuming the continuum hypothesis, that for every $\alpha \in [0,1]$ there is a subfield of $\mathbb{R}$, hence also a subring, whose Hausdorff dimension is exactly $\alpha$; its abstract describes the proof as the completion of an approach first found by Roy Davies. The endpoints are trivial ($\mathbb{Q}$ has dimension $0$ and $\mathbb{R}$ has dimension $1$); the content is the open interval.

Covers. One side of Problem 1154, that ZFC does not disprove the existence: a model of ZFC in which the continuum hypothesis holds contains, for every $\alpha \in [0,1]$, a subfield of dimension $\alpha$. Since the continuum hypothesis is consistent with ZFC when ZFC is consistent, no proof in ZFC can refute the existence the problem asks about, so the question is not disprovable in ZFC. That side alone leaves the question open: the theorem does not decide whether ZFC alone proves the existence, and only a matching result that ZFC does not prove it would settle the problem, as independent. It does not conflict with the Edgar-Miller theorem that a Borel or analytic subring of $\mathbb{R}$ has dimension $0$ or is all of $\mathbb{R}$: any field of intermediate dimension is necessarily non-analytic.

Acceptance. The result is refereed: it appeared in Mathematical Proceedings of the Cambridge Philosophical Society. The site's curator, Thomas Bloom, labels Problem 1154 not disprovable and credits the label to this theorem in the problem's remarks; a thread comment of 2026-01-24 pointed to the paper. That independent credit is the reviewed evidence. The corpus has not reproved the theorem and no formalization is recorded.