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Measurable circle squaring

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<mark>Journal publication date</mark>03/2017
<mark>Journal</mark>Annals of Mathematics
Issue number2
Volume185
Number of pages40
Pages (from-to)671-710
Publication StatusPublished
Early online date1/03/17
<mark>Original language</mark>English

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.

Bibliographic note

40 pages; Lemma 4.4 improved & more details added; accepted by Annals of Mathematics