Measurable circle squaring

Abstract

Laczkovich proved that if bounded subsets A and B of R k have the same non-zero 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.

Locations

• arXiv (Cornell University) - View - PDF
• Warwick Research Archive Portal (University of Warwick) - View - PDF
• Lancaster EPrints (Lancaster University) - View - PDF
• Annals of Mathematics - View