Geodesic
Measurable circle squaring
Łukasz Grabowski1 ; András Máthé2 ; Oleg Pikhurko3
1 Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom
2 Mathematics Institute, University of Warwick, Coventry, United Kingdom
3 Mathematics Institute and DIMAP, University of Warwick, Coventry, United Kingdom
Annals of mathematics, Tome 185 (2017) no. 2, pp. 671-710.

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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.
DOI : 10.4007/annals.2017.185.2.6

Łukasz Grabowski 1 ; András Máthé 2 ; Oleg Pikhurko 3

1 Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom
2 Mathematics Institute, University of Warwick, Coventry, United Kingdom
3 Mathematics Institute and DIMAP, University of Warwick, Coventry, United Kingdom 
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Łukasz Grabowski; András Máthé; Oleg Pikhurko. Measurable circle squaring. Annals of mathematics, Tome 185 (2017) no. 2, pp. 671-710. doi : 10.4007/annals.2017.185.2.6. http://geodesic.mathdoc.fr/articles/10.4007/annals.2017.185.2.6/

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