On the Hausdorff dimension of microsets

TL;DR

This paper explores the relationship between the Hausdorff dimensions of microsets and the original set, establishing that the minimal and maximal microset dimensions correspond to the lower and Assouad dimensions, respectively.

Contribution

It proves that the lower dimension equals the minimal microset dimension and constructs sets with microset dimensions exactly matching any given closed interval.

Findings

Maximal microset dimension equals the Assouad dimension.

Minimal microset dimension equals the lower dimension.

Existence of sets with microset dimensions forming any specified closed interval.

Abstract

We investigate how the Hausdorff dimensions of microsets are related to the dimensions of the original set. It is known that the maximal dimension of a microset is the Assouad dimension of the set. We prove that the lower dimension can analogously be obtained as the minimal dimension of a microset. In particular, the maximum and minimum exist. We also show that for an arbitrary $\mathcal{F}_\sigma$ set $\Delta \subseteq [0,d]$ containing its infimum and supremum there is a compact set in $[0,1]^d$ for which the set of Hausdorff dimensions attained by its microsets is exactly equal to the set $\Delta$. Our work is motivated by the general programme of determining what geometric information about a set can be determined at the level of tangents.