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

The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any $0<β<α$, any compact metric space $X$ of Hausdorff dimension $α$ contains a subset which is biLipschitz equivalent to an ultrametric and has Hausdorff dimension at least $β$. 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.

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