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arXiv · 2104.01244

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

Tomasz Cieśla·Łukasz Grabowski
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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.

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Tomasz Cieśla, Łukasz Grabowski. 2021-04-02. On random compact sets, equidecomposition, and domains of expansion in R^3. https://arxiv.org/abs/2104.01244

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