Dvoretzky-type theorem for Ahlfors regular spaces

TL;DR

This paper proves that bounded Ahlfors -regular spaces contain -regular subsets that embed into ultrametrics with controlled distortion, extending Dvoretzky-type theorems to metric measure spaces.

Contribution

It introduces a new embedding result for Ahlfors regular spaces into ultrametrics with explicit distortion bounds, using a regular ultrametric skeleton theorem.

Findings

Existence of -regular subsets embedding into ultrametrics with distortion O(/(-)))

Distortion bounds are asymptotically tight as  approaches 

Extension of Dvoretzky-type theorems to Ahlfors regular metric 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.