On random compact sets, equidecomposition and domains of expanison in R^3

We describe a certain model of random compact sets. We use it to construct a pair of compact sets A and B in R^3 of the same Lebesgue measure such that A can be covered by finitely many translates of B and vice versa, yet A and B are not equidecomposable. We also give the first example of a compact subset of R^3 of positive measure which is not a domain of expansion. This is joint work with Łukasz Grabowski.