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.

中文翻译：

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倍空间中 Hausdorff 维数的 Dvoretzky 型定理的简单证明

超量规骨架定理 [Mendel, Naor 2013] 暗示了 Hausdorff 维数的以下非线性 Dvoretzky 型定理： 对于任何 0 < β < α，任何紧凑度量空间XHausdorff 维度 α 的子集是 biLipschitz 等价于超度量并且 Hausdorff 维度至少为 β。在本笔记中，我们使用 Bartal 的 Ramsey 分解 [Bartal 2021] 提出了在倍增空间中超量测骨架定理的简单证明。同样的一般方法也用于回答 Zindulka [Zindulka 2020] 关于是否存在具有完整 Hausdorff 维度的紧凑空间的“近乎超度量”子集的问题。