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

IF 0.9 3区数学 Q2 MATHEMATICS

Analysis and Geometry in Metric Spaces Pub Date : 2021-04-24 DOI:10.1515/agms-2022-0133

M. Mendel

引用次数: 1

Abstract

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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重倍空间中Hausdorff维dvoretzky型定理的一个简单证明

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

16 weeks

期刊介绍： Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.