On the uniformity and size of microsets

TL;DR

This paper investigates the properties of microsets in Euclidean space, demonstrating the existence of sets with near-maximal Assouad dimension whose microsets lack high-dimensional regular subsets, and showing that microsets can have finite packing pre-measure.

Contribution

It constructs sets with high Assouad dimension whose microsets lack high-dimensional regular subsets and proves the existence of microsets with finite packing pre-measure for any set with positive lower dimension.

Findings

Constructed a compact set with Assouad dimension close to d with microsets lacking high-dimensional regular subsets.

Proved existence of microsets with finite β-dimensional packing pre-measure for sets with lower dimension β.

Answered open questions regarding the uniformity and size of microsets in Euclidean space.

Abstract

We resolve a few questions regarding the uniformity and size of microsets of subsets of Euclidean space. First, we construct a compact set $K\subset\mathbb{R}^d$ with Assouad dimension arbitrarily close to $d$ such that every microset of $K$ has no Ahlfors--David regular subset with dimension strictly larger than $0$. This answers a question of Orponen. Then, we show that for any non-empty compact set $K\subset\mathbb{R}^d$ with lower dimension $\beta$, there is a microset $E$ of $K$ with finite $\beta$-dimensional packing pre-measure. This answers a strong version of a question of Fraser--Howroyd--K\"aenm\"aki--Yu, who previously obtained a similar result concerning the upper box dimension.