Gardner conjectured that if two bounded measurable sets A, B ⊂ ℝⁿ are equidecomposable by a set of isometries Γ 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 Γ. We answer this question negatively.

中文翻译：

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关于加德纳猜想

Gardner 猜想，如果两个有界可测集A, B ⊂ ℝ ⁿ可被一组等距Γ^等式分解，从而生成一个适合的组，则A和B允许所有等距的可测等分。Cieśla 和 Sabok 询问是否存在仅在Γ生成的组中使用等距的可测量等量分解。我们对这个问题的回答是否定的。