Dvoretzky-type theorem for Ahlfors regular spaces

Abstract

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