On the Hausdorff spectra of free pro-$p$ groups and certain $p$-adic analytic groups

We establish that finitely generated non-abelian direct products $G$ of free pro-$p$ groups have full Hausdorff spectrum with respect to the lower $p$-series $\mathcal{L}$. This complements similar results with respect to other standard filtration series and a recent theorem showing that the Hausdorff spectrum ${\text{hspec}}^{\mathcal{L}}(G)$ of a $p$-adic analytic pro-$p$ group $G$ is discrete and consists of at most $2^{\dim (G)}$ rational numbers. The latter also left some room for improvement regarding the upper bound. Indeed, for finitely generated nilpotent pro-$p$ groups $G$ we obtain the stronger assertion that the cardinality of the Hausdorff spectrum is at most the analytic dimension of $G$. Moreover, we produce a corresponding result when the $p$-adic analytic pro-$p$ group $G$ is just infinite, which holds not just for the lower $p$-series but for arbitrary filtration series. Finally, we show that, if $G$ is a countably based pro-$p$ group with an open subgroup mapping onto the free abelian pro-$p$ group ${\mathbb{Z}}_p\oplus {\mathbb{Z}}_p$, then for every prescribed finite set $\{0,1\}\subseteq X\subseteq [0,1]$ there is a filtration series $\mathcal{S}$ such that ${\text{hspec}}^{\mathcal{S}}(G)=X$; in particular, $\mathrm{\mid }{\text{hspec}}^{\mathcal{S}}(G)\mathrm{\mid }$ is unbounded, as $\mathcal{S}$ runs through all filtration series of $G$ with $\mathrm{\mid }{\text{hspec}}^{\mathcal{S}}(G)\mathrm{\mid }<\mathrm{\infty }$.