Assouad dimension, Nagata dimension, and uniformly close metric tangents

TL;DR

This paper investigates the relationship between Assouad and Nagata dimensions in metric spaces, establishing bounds and conditions under which these local dimensions match tangent space dimensions, with applications to subRiemannian manifolds.

Contribution

It proves that Nagata dimension is always bounded by Assouad dimension and identifies conditions for local dimensions to match tangent space dimensions.

Findings

Nagata dimension is bounded above by Assouad dimension.

Uniformly close tangents alone do not determine local dimensions.

In equiregular subRiemannian manifolds, Nagata dimension equals topological dimension.

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 subRiemannian manifolds and show that locally their Nagata dimension equals the topological dimension.