On Gardner's conjecture

2021/03/22 by Kun, Gabor
#Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2103.11655

Abstract

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.

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