$G_\delta$ Circle Squaring

TL;DR

This paper proves that a circle and a square of equal area in the plane can be dissected into finitely many pieces that are both $F_\sigma$ and $G_\delta$, improving previous results and achieving optimal Borel complexity.

Contribution

It establishes the equidecomposability of plane shapes with minimal Borel complexity, extending prior work and introducing new low-complexity constructions in Borel combinatorics.

Findings

Circle and square of equal area are $F_\sigma$ and $G_\delta$ equidecomposable.

Generalization to bounded sets with small boundary and same measure.

Construction of low complexity toasts for Borel combinatorics.

Abstract

We show that a circle and square of the same area in $\mathbb{R}^2$ are equidecomposable by translations using $\mathbf{\Delta}^0_2$ pieces. That is, pieces which are simultaneously $F_\sigma$ and $G_\delta$ sets. This improves a result of M\'ath\'e-Noel-Pikhurko and is the best possible complexity in terms of the Borel hierarchy. More generally we show that bounded sets $A,B \subseteq \mathbb{R}^n$ with small enough boundaries and the same nonzero Lebesgue measure are equidecomposable with pieces that are countable unions of finite Boolean combinations of translates of $A,B$, and open sets. The improvement comes from constructions of low complexity toasts and related objects which should be independently useful within Borel combinatorics.