Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The range of dimensions of microsets   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 a 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.

27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th