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

TL;DR

This paper provides a straightforward proof of a Dvoretzky-type theorem for Hausdorff dimension in doubling spaces, showing the existence of large ultrametric-like subsets within compact metric spaces.

Contribution

It offers a simple proof of the ultrametric skeleton theorem in doubling spaces using Bartal's Ramsey decompositions, and addresses the existence of nearly ultrametric subsets with full Hausdorff dimension.

Findings

Existence of ultrametric subsets with controlled Hausdorff dimension

Simplified proof of the ultrametric skeleton theorem in doubling spaces

Answer to Zindulka's question on nearly ultrametric subsets

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.