Dvoretzky-type theorem for Ahlfors regular spaces

It is proved that for any $0<\beta<\alpha$, any bounded Ahlfors $\alpha$-regular space contains a $\beta$-regular compact subset that embeds biLipschitzly in an ultrametric with distortion at most $O(\alpha/(\alpha-\beta))$. The bound on the distortion is asymptotically tight when $\beta\to \alpha$. The main tool used in the proof is a regular form of the ultrametric skeleton theorem.