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

Abstract

The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any $0<\beta <\alpha$, any compact metric space $X$ of Hausdorff dimension $\alpha$ contains a subset which is biLipschitz equivalent to an ultrametric and has Hausdorff dimension at least $\beta$. In this note we present a simple proof of the ultrametric skeleton theorem in doubling spaces using Bartal's Ramsey decompositions [Bartal 2021]. The same general approach is also used to answer a question of Zindulka [Zindulka 2020] about the existence of "nearly ultrametric" subsets of compact spaces having full Hausdorff dimension.

13 pages, 1 figure. Minor changes and improvements. Accepted to "Analysis and Geometry in Metric Spaces"

References in corpus (2)

• Advances in Metric Ramsey Theory and its Applications
• Dvoretzky-type theorem for Ahlfors regular spaces

Cited by in corpus (1)

• Dvoretzky-type theorem for Ahlfors regular spaces