$G_δ$ Circle Squaring

TL;DR

This paper proves that a circle and a square of equal area in $\mathbb{R}^2$ are equidecomposable by translations using $\mathbf{\Delta}^0_2$ pieces, establishing the best possible Borel complexity. The authors develop low-complexity toasts on Schreier graphs via asymptotic-dimension witnesses and construct bounded geometry decompositions to enable precise combinatorial control. They generalize circle-squaring-type results to bounded sets in $\mathbb{R}^k$ with small boundary and equal positive measure, showing equidecomposability with pieces that are countable unions of Boolean combinations of translates and open sets. The methods yield an effectively constructive pathway (via a rounding algorithm) to explicit equidecompositions, and they highlight the broader utility of low-complexity toasts in descriptive combinatorics, with potential for further resolutions in the Borel hierarchy and related hierarchies.

Abstract

We show that a circle and square of the same area in $\mathbb{R}^2$ are equidecomposable by translations using $\mathbfΔ^0_2$ pieces. That is, pieces which are simultaneously $F_σ$ and $G_δ$ sets. This improves a result of Máthé-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.

Theorems & Definitions (66)

• Theorem 1.1
• Definition 1.3
• Theorem 1.4
• Lemma 1.5
• proof
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8: GJKS
• Theorem 1.9
• Definition 2.1