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September 2017 Borel circle squaring
Andrew S. Marks, Spencer T. Unger
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Ann. of Math. (2) 186(2): 581-605 (September 2017). DOI: 10.4007/annals.2017.186.2.4

Abstract

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 $k\ge 1$ and $A,B\subseteq \mathbb{R}^k$ 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 $\mathbb{Z}^d$.

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Andrew S. Marks. Spencer T. Unger. "Borel circle squaring." Ann. of Math. (2) 186 (2) 581 - 605, September 2017. https://doi.org/10.4007/annals.2017.186.2.4

Information

Published: September 2017
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2017.186.2.4

Subjects:
Primary: 03E15 , 05C21 , 37A20 , 52B45

Keywords: amenability , circle squaring , descriptive graph combinatorics , descriptive set theory , Flows , hyperfiniteness

Rights: Copyright © 2017 Department of Mathematics, Princeton University

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Vol.186 • No. 2 • September 2017
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