The range of dimensions of microsets

Abstract

We say that $E$ is a microset of the compact set $K\subset {\mathbb{R}}^d$ if there exist sequences ${\lambda }_n\ge 1$ 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, $E\cap (0,1{)}^d\ne \varnothing$. 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 a question of Fraser, Howroyd, Käenmäki, 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.

21 pages, the proofs of Theorems 4.7 and 4.8 were improved, also some minor modifications