On Gardner's conjecture

Abstract

Gardner conjectured that if two bounded measurable sets $A,B\subset {\mathbb{R}}^n$ are equidecomposable by a set of isometries $\Gamma$ generating an amenable group then $A$ and $B$ admit a measurable equidecomposition by all isometries. Cieśla and Sabok asked if there is a measurable equidecomposition using isometries only in the group generated by $\Gamma$. We answer this question negatively.

The proofs have simplified. It will appear in Combinatorica