Borel Circle Squaring

TL;DR

This paper provides a constructive Borel solution to Tarski's circle squaring problem, proving that certain bounded Borel sets with equal measure are equidecomposable by translations with Borel pieces.

Contribution

It establishes a Borel version of Laczkovich's equidecomposition theorem, answering Wagon's question and extending the scope of measurable equidecompositions.

Findings

Proves Borel equidecomposability for sets with boundary Minkowski dimension less than the ambient space.

Uses graph flow techniques and recent hyperfiniteness results in Borel dynamics.

Provides a constructive approach to classical geometric measure theory problems.

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 $\mathbb{Z}^d$.