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.
Bibliographical notePublisher Copyright:
© 2022 Manor Mendel, published by De Gruyter.
- biLipschitz embeddings
- Dvoretzky-type theorems
- Hausdorff dimension
- Metric Ramsey theory