Assouad Dimension, Nagata Dimension, and Uniformly Close Metric Tangents

Abstract

We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the dimensions of its metric tangents. Having uniformly close tangents is not sufficient. What is needed, in addition, is either that the tangents have dimension with uniform constants independent from the point and the tangent, or that the tangents are unique. We will apply our results to equiregular sub-Riemannian manifolds and show that, locally, their Nagata dimension equals the topological dimension.