Measurable Hall's theorem for actions of abelian groups

2019/03/07 by Cieśla, Tomasz, Sabok, Marcin · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1903.02987

Abstract

We prove a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. In particular, it implies that for free measure-preserving actions of such groups, if two equidistributed measurable sets are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner.

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• Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability

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