Hausdorff Dimensions of Computable Closed Subgroups of Finitely Generated Profinite Groups

In a 2025 list of open problems in computability theory, A. Nies notes that every left-\(\Sigma_2\) real \(r\in[0,1]\) is the Hausdorff dimension of a computable subgroup of a computable profinite group, and asks which values occur when the group is topologically finitely generated. Reading “computable subgroup” as computable closed subgroup, and taking the dimension, as Mayordomo and Nies do, with respect to the given computable inverse system, we show that finite generation imposes no restriction: the values are exactly the left-\(\Sigma_2\) reals in \([0,1]\). All of them occur for computable closed subgroups of the \(2\)-generated pro-\(p\) lamplighter group \(\mathbb{Z}_p\ltimes\mathbb{F}_p[\![t]\!]\), simultaneously for its presentation by the wreath products \(C_p\wr C_{p^k}\) and for its \(p\)-power, iterated \(p\)-power, Frattini and dimension subgroup series; for its lower \(p\)-series the values are the left-\(\Sigma_2\) reals in \([0,\frac12]\), the numbers \(\frac12+\frac m{2p^n}\) with \(0<m<p^n\), and \(1\). If “computable subgroup” meant “closed subgroup generated by finitely many computable elements”, the question would become an effective version of the problem of finitely generated Hausdorff spectra, which we do not address. The question and the statement without finite generation are from Nies's section of the list; the statement is given there without proof and follows from an example of Mayordomo and Nies (essentially due to Barnea and Shalev) and the theorem of Downey, Jockusch and Schupp that the lower densities of computable sets are exactly the left-\(\Sigma_2\) reals. The Hausdorff spectra of this group are classical (Klopsch; Klopsch and Thillaisundaram; de las Heras and Klopsch). The note is a straightforward effective version of these results, built on a lemma that makes one computable set give the prescribed dimension along several filtrations at once; its novelty is modest. This is an unrefereed note.