  We study random compact subsets of R^3 which can be described as "random
Menger sponges". We use those random sets to construct a pair of compact sets A
and B in R^3 which are of the same positive measure, such that A can be covered
by finitely many translates of B, B can be covered by finitely many translates
of A, and yet A and B are not equidecomposable. Furthermore, we construct the
first example of a compact subset of R^3 of positive measure which is not a
domain of expansion. This answers a question of Adrian Ioana.
