TY - JOUR
T1 - A Simple Proof of Dvoretzky-Type Theorem for Hausdorff Dimension in Doubling Spaces
AU - Mendel, Manor
N1 - Publisher Copyright:
© 2022 Manor Mendel, published by De Gruyter.
PY - 2022/1/1
Y1 - 2022/1/1
N2 - 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.
AB - 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.
KW - Dvoretzky-type theorems
KW - Hausdorff dimension
KW - Metric Ramsey theory
KW - biLipschitz embeddings
UR - http://www.scopus.com/inward/record.url?scp=85129274353&partnerID=8YFLogxK
U2 - 10.1515/agms-2022-0133
DO - 10.1515/agms-2022-0133
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AN - SCOPUS:85129274353
SN - 2299-3274
VL - 10
SP - 50
EP - 62
JO - Analysis and Geometry in Metric Spaces
JF - Analysis and Geometry in Metric Spaces
IS - 1
ER -