Erdős Problem 1154 on Hausdorff dimensions of subrings and subfields

Determine whether, for every real number \(\alpha\in[0,1]\), there exists a subring or subfield \(F\subseteq\mathbb{R}\) whose Hausdorff dimension is \(\alpha\).

Background

Erdős Problem 1154 asks whether every value in the interval $[0,1]$ can occur as the Hausdorff dimension of a subring or subfield of the real numbers. The paper contrasts this question with the known result that every value in $[0,1]$ occurs as the Hausdorff dimension of an additive subgroup of $\mathbb{R}$, while multiplication imposes substantially stronger structural restrictions.

Previous results cited in the paper show that analytic subrings satisfy a zero–one law, whereas, assuming the Continuum Hypothesis, subfields of every prescribed Hausdorff dimension can be constructed. Thus the general existence question for arbitrary subrings or subfields remains the explicitly stated problem motivating the paper, although the paper’s main theorem establishes a zero–one law for subfields arising from Turing ideals rather than resolving Erdős Problem 1154 in full.