Measurable circle squaring

Grabowski, Łukasz and Máthé, András and Pikhurko, Oleg (2017) Measurable circle squaring. Annals of Mathematics, 185 (2). pp. 671-710. ISSN 0003-486X

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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.