Borel Circle Squaring

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 \(k \geq 1\) 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 \(\mathbb{Z}^d\).