Dvoretzky-type theorem for Ahlfors regular spaces

Abstract

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.

20 pages. Minor editing. Accepted to Studia Mathematica

References in corpus (5)

• Fast Construction of Nets in Low Dimensional Metrics, and Their Applications
• Ahlfors-David regular sets and bilipschitz maps
• Packing dimension and Ahlfors regularity of porous sets in metric spaces
• Advances in Metric Ramsey Theory and its Applications
• A simple proof of Dvoretzky-type theorem for Hausdorff dimension in doubling spaces

Cited by in corpus (1)

• A simple proof of Dvoretzky-type theorem for Hausdorff dimension in doubling spaces