重倍空间中Hausdorff维dvoretzky型定理的一个简单证明

Pub Date : 2021-04-24 DOI:10.1515/agms-2022-0133

M. Mendel

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引用次数: 1

摘要

摘要超度量骨架定理[Mendel, Naor 2013]推导出以下Hausdorff维数的非线性dvoretzky型定理:对于任意0 < β < α，任意Hausdorff维数α的紧度量空间X包含一个与超度量等价且Hausdorff维数至少为β的biLipschitz子集。在这篇文章中，我们使用Bartal的Ramsey分解给出了在加倍空间中超度量骨架定理的一个简单证明[Bartal 2021]。同样的一般方法也用于回答Zindulka [Zindulka 2020]关于具有完整Hausdorff维的紧化空间的“近超度量”子集的存在性的问题。

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

Abstract 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.

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