Gardner's measurable equidecomposition conjecture

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Gardner's measurable equidecomposition conjecture

   Let $A,B\subseteq\mathbb{R}^n$ be bounded measurable sets, and let $\Gamma$ be an amenable group of isometries. Two sets are $\Gamma$-equidecomposable if they admit finite partitions whose corresponding pieces are mapped to one another by elements of $\Gamma$; an equidecomposition is measurable if all partition pieces are measurable. Gardner's conjecture. If $A$ and $B$ are $\Gamma$-equidecomposable, then they admit a measurable equidecomposition. This conjecture asks when equidecompositions by amenable groups can be realized measurably; the paper studies this question and proves a characterization for amenable groups in terms of finite-subgroup quotients, while the general formulation here is presented as Gardner's conjecture.

   source: Matthew Bowen, Gábor Kun and Marcin Sabok, “The uniform Gardner conjecture and rounding Borel flows”, arXiv:2212.03785 (2022).