Measurable circle squaring

2015/01/25 by Grabowski, Łukasz, Máthé, András, Pikhurko, Oleg · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1501.06122

Abstract

Laczkovich proved that if bounded subsets A and B of R^k have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than k, then there is a partition of A into finitely many parts that can be translated to form a partition of B. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-only versions of Tarski's circle squaring and Hilbert's third problem.

Cited by

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

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• Circle Squaring with Pieces of Small Boundary and Low Borel Complexity
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