A Simple Proof of Dvoretzky-Type Theorem for Hausdorff Dimension in Doubling Spaces

Analysis and Geometry in Metric Spaces

Mendel Manor¹ 

[1] Mathematics and Computer Science Department, The Open University of Israel, 1 University Road, P.O. Box 808, Raanana 43107, Israel;

关键词: hausdorff dimension;    metric ramsey theory;    bilipschitz embeddings;    dvoretzky-type theorems;    51f30;    28a78;    46b85;   

DOI  :  10.1515/agms-2022-0133

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown