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.

arXiv comment20 pages. Minor editing. Accepted to Studia Mathematica

References in corpus (8)

• Fast Construction of Nets in Low Dimensional Metrics, and Their Applications
• On metric Ramsey-type phenomena
• Ahlfors-David regular sets and bilipschitz maps
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• Ultrametric skeletons
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• Ahlfors regular spaces have regular subspaces of any dimension
• 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