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 $k\ge 1$ 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 ${\mathbb{Z}}^d$.

Minor typos corrected

Cited by in corpus (3)

• Circle Squaring with Pieces of Small Boundary and Low Borel Complexity
• Minimal definable graphs of definable chromatic number at least three
• Measurable realizations of abstract systems of congruences