Approximating spaces of Nagata dimension zero by weighted trees

TL;DR

This paper demonstrates that metric spaces with Nagata dimension zero can be densely approximated by weighted trees, providing new insights into their structure and embedding properties, with implications for Lipschitz extensions and hyperbolic spaces.

Contribution

It establishes that Nagata dimension zero spaces are densely approximable by weighted trees, extending previous results and applying to hyperbolic spaces.

Findings

Dense subsets are bilipschitz equivalent to weighted trees

Best possible factor of 8 for ultrametric spaces

Quantitative metric embedding and Lipschitz extension results

Abstract

We prove that if a metric space $X$ has Nagata dimension zero with constant $c$, then there exists a dense subset of $X$ that is $8c$-bilipschitz equivalent to a weighted tree. The factor $8$ is the best possible if $c=1$, that is, if $X$ is an ultrametric space. This yields a new proof of a result of Chan, Xia, Konjevod and Richa. Moreover, as an application, we also obtain quantitative versions of certain metric embedding and Lipschitz extension results of Lang and Schlichenmaier. Finally, we prove a variant of our main theorem for $0$-hyperbolic proper metric spaces. This generalizes a result of Gupta.