Standard Hausdorff spectrum of compact $$\mathbb {F}_p[[t]]$$-analytic groups

The finitely generated Hausdorff spectra of a family of pro-p groups

“…These open normal subgroups yield a base of neighbourhoods of the identity and induce a translation-invariant metric on Γ which is given by d S (x, y) = inf |Γ : Γ i | −1 | x ≡ y (mod Γ i ) , for x, y ∈ Γ. This gives, for a subset U ⊆ Γ, the Hausdorff dimension hdim S Γ (U ) ∈ [0, 1] with respect to the filtration series S. Recently there has been much interest concerning Hausdorff dimensions in profinite groups, starting with the pioneering work of Abercrombie [1] and of Barnea and Shalev [3]; recent work includes for example [2,4,[7][8][9][10][12][13][14]16,18]. Barnea and Shalev [3] proved the following group-theoretic formula of the Hausdorff dimension with respect to S of a closed subgroup H of Γ as a logarithmic density:…”