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

Manor Mendel
2022 arXiv   pre-print
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
more » ... 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.
arXiv:2104.11944v4 fatcat:6fkzrdhkjrf2ffu2uux3ervxlm