On random compact sets, equidecomposition, and domains of expansion in R^3
Metric Geometry
2021-04-06 v1 Combinatorics
Logic
Probability
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.
Keywords
Cite
@article{arxiv.2104.01244,
title = {On random compact sets, equidecomposition, and domains of expansion in R^3},
author = {Tomasz Cieśla and Łukasz Grabowski},
journal= {arXiv preprint arXiv:2104.01244},
year = {2021}
}
Comments
15 pages