On Gardner's conjecture

TL;DR

This paper disproves a conjecture by Gardner, showing that two equidecomposable sets under an amenable group cannot always be equidecomposed using only the isometries in the generated subgroup.

Contribution

It provides a negative answer to a question about measurable equidecompositions restricted to subgroup isometries, clarifying limitations in Gardner's conjecture.

Findings

Disproved Gardner's conjecture in specific cases.

Showed limitations of subgroup isometries in measurable equidecompositions.

Established that not all equidecompositions can be realized within the generated subgroup.

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. Cie\'sla and Sabok asked if there is a measurable equidecomposition using isometries only in the group generated by $\Gamma$. We answer this question negatively.