Effective Hausdorff dimensions in a two-generated profinite group

We classify the Hausdorff dimensions of computable closed subgroups of computable profinite groups: the possible values are exactly the left-\(\Sigma^0_2\) reals in \([0,1]\).  A single computable two-generated pro-\(p\) group suffices for every value.  The construction computes its intrinsic power filtration and resamples computable lower densities at exponentially sparse filtration levels.

This manuscript addresses the intended mathematical problem identified in the source.

AI disclosure: This problem was solved using GPT 5.6 Sol.

Mathematics subject: math.LO; source problem: OP-0751.