Approximating spaces of Nagata dimension zero by weighted trees

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 $8c$-bi-Lipschitz 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.