Geodesic
Borel circle squaring
Andrew S. Marks1 ; Spencer T. Unger1
1 University of California at Los Angeles, Los Angeles, CA
Annals of mathematics, Tome 186 (2017) no. 2, pp. 581-605.

Voir la notice de l'article provenant de la source Annals of Mathematics website

We give a completely constructive solution to Tarski’s circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subset \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.
DOI : 10.4007/annals.2017.186.2.4

Andrew S. Marks 1 ; Spencer T. Unger 1

1 University of California at Los Angeles, Los Angeles, CA 
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Andrew S. Marks; Spencer T. Unger. Borel circle squaring. Annals of mathematics, Tome 186 (2017) no. 2, pp. 581-605. doi : 10.4007/annals.2017.186.2.4. http://geodesic.mathdoc.fr/articles/10.4007/annals.2017.186.2.4/

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