Measurable circle squaring

Abstract

Laczkovich proved that if bounded subsets $A$ and $B$ of $\mathbb{R}^k$ have the same nonzero Lebesgue measure and the upper 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.