Measurable circle squaring

TL;DR

This paper extends classical circle squaring results by ensuring the parts are both Baire and Lebesgue measurable, providing new measurable versions of Tarski's circle squaring and Hilbert's third problem.

Contribution

It proves that partitions in circle squaring can be made with parts that are both Baire and Lebesgue measurable, enhancing previous results that lacked this measurability condition.

Findings

Partitions can be made with measurable parts

Measurable circle squaring versions of Tarski's problem

Measurable solutions to Hilbert's third problem

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.