It is proved that for any 0 < β < α, any bounded Ahlfors α-regular space contains a β-regular compact subset that embeds biLipschitzly in an ultrametric with distortion at most O(α/(α − β)). The bound on the distortion is asymptotically tight when β → α. The main tool used in the proof is a regular form of the ultrametric skeleton theorem.
Bibliographical noteFunding Information:
a preliminary version of the paper. The author also acknowledges support by U.S.-Israel Binational Science Foundation, grant 2018223.
© Instytut Matematyczny PAN, 2023.
- Ahlfors regular spaces
- biLipschitz embeddings
- Dvoretzky-type theorems
- Hausdorff dimension
- metric Ramsey theory
- ultrametric skeleton