On the uniformity and size of microsets

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--Käenmäki--Yu, who previously obtained a similar result concerning the upper box dimension.

arXiv comment15 pages. v2: Minor typo fixes. To appear in Proc RSE A