On the uniformity and size of microsets
Richárd Balka, Vilma Orgoványi, Alex Rutar
TL;DR
The paper tackles how uniform microset regularity is and how large microsets can be in Euclidean spaces. It develops a dyadic and generalized-cube framework to analyze microsets and constructs a uniformly branching set \(K(\bm{a})\) to force dimensional controls, showing that positive-dimensional Ahlfors–David regular subsets need not appear in microsets even when \(\dim_H K=\dim_A K=\alpha\) with \(0\le\alpha<d\). It also proves that for any compact \(K\) with lower dimension \(\beta\), there exists a tangent \(F\in\mathrm{Tan}(K)\) with finite packing pre-measure \(\mathcal{P}_0^{\beta}(F)<\infty\) (indeed with an explicit bound on \(\mathcal{P}^\beta(F)\)). These results refine our understanding of the regularity of microsets, demonstrate sharp limitations on finding regular subsets within all microsets, and introduce robust, tree-based methods for controlling packing measures of tangents, with potential implications for projection phenomena and dimension theory in fractal geometry.
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 $β$, there is a microset $E$ of $K$ with finite $β$-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.