Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 1-22 |
| Number of pages | 22 |
| Journal | Studia Mathematica |
| Volume | 268 |
| Issue number | 1 |
| DOIs | |
| State | Published - 2023 |
Bibliographical note
Funding Information:a preliminary version of the paper. The author also acknowledges support by U.S.-Israel Binational Science Foundation, grant 2018223.
Publisher Copyright:
© Instytut Matematyczny PAN, 2023.
Keywords
- Ahlfors regular spaces
- biLipschitz embeddings
- Dvoretzky-type theorems
- Hausdorff dimension
- metric Ramsey theory
- ultrametric skeleton