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

TL;DR

This paper constructs examples of compact sets in R^3 with surprising properties, including non-equidecomposability despite measure equivalence, and identifies a set that is not a domain of expansion, answering longstanding questions.

Contribution

It introduces the first examples of non-equidecomposable sets of equal measure and a set that is not a domain of expansion in R^3, advancing understanding of geometric measure theory.

Findings

Constructed pairs of sets with equal measure but not equidecomposable.

Provided the first example of a set not being a domain of expansion in R^3.

Used random Menger sponges to build these examples.

Abstract

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.