The range of dimensions of microsets

Richárd Balka, Márton Elekes, Viktor Kiss

Published 2021-02-25Version 1

We say that $E$ is a microset of the compact set $K\subset \mathbb{R}^d$ if there exist sequences $\lambda_n\geq 1$ and $u_n\in \mathbb{R}^d$ such that $(\lambda_n K + u_n ) \cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and moreover, $E \cap (0, 1)^d \neq \emptyset$. The main result of the paper is that for a non-empty set $A\subset [0,d]$ there is a compact set $K\subset \mathbb{R}^d$ such that the set of Hausdorff dimensions attained by the microsets of $K$ equals $A$ if and only if $A$ is analytic and contains its infimum and supremum. This answers another question of Fraser, Howroyd, K\"aenm\"aki, and Yu. We show that for every compact set $K\subset \mathbb{R}^d$ and non-empty analytic set $A\subset [0,\dim_H K]$ there is a set $\mathcal{C}$ of compact subsets of $K$ which is compact in the Hausdorff metric and $\{\dim_H C: C\in \mathcal{C} \}=A$. The proof relies on the technique of stochastic co-dimension applied for a suitable coupling of fractal percolations with generation dependent retention probabilities. We also examine the analogous problems for packing and box dimensions.