Coanalytic subfields of the reals of every Hausdorff dimension

An analytic subring of $\mathbb{R}$ has Hausdorff dimension $0$ or is equal to $\mathbb{R}$, by a theorem of Edgar–Miller and Bourgain. We show that this fails for coanalytic subfields in the constructible universe: if $V=L$, then for every $\alpha \in [0,1]$ there is a proper coanalytic subfield of $\mathbb{R}$ of Hausdorff dimension $\alpha$. In fact there is a single parameter-free ${\mathrm{\Pi }}_1^1$ set $G\subseteq [0,1]\times \mathbb{R}$ whose sections $A(\alpha )$ are such fields, are strictly increasing in $\alpha$, and satisfy $A(\alpha )={\bigcup }_{r<\alpha }A(r)$ for $\alpha >0$.