Measurable circle squaring


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.


Łukasz Grabowski

Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom

András Máthé

Mathematics Institute, University of Warwick, Coventry, United Kingdom

Oleg Pikhurko

Mathematics Institute and DIMAP, University of Warwick, Coventry, United Kingdom