Ultrametric spaces and the logarithmic ratio

It is shown that if a compact metric space $(X,d)$ is bi-Hölder equivalent to an ultrametric space, then the logarithmic ratio $R(X,d)$ is finite. Conversely, if the logarithmic ratio $R(X,d)$ is finite and ${\text{A}}_p^{\ast }(X)\mathrm{\ne }\mathrm{\varnothing }$ for some $p\in (1,\mathrm{\infty })$, then $(X,d)$ is bi-Hölder equivalent to an ultrametric space. It is also shown that for any $s\in [0,\mathrm{\infty }]$ there exists a compact countable metric space $(X,d)$ with a unique cluster point such that the logarithmic ratio $R(X,d)$ is equal $s$. Moreover, we prove a bi-Hölder embedding result for a certain class of compact totally disconnected metric spaces.