G_δ Circle Squaring

2026/01/26 by Spencer Unger, Narmada Varadarajan, Felix Weilacher · 1 voice
Mathematics · #math.LO #math.CO #math.MG

paper · pdf

arxiv published 2026/01/26 · arxiv updated 2026/02/25

Abstract

We show that a circle and square of the same area in ℝ² are equidecomposable by translations using \mathbfΔ⁰₂ 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 ⊆ ℝⁿ 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.

Citations

• Borel Combinatorics of Abelian Group Actions
• Circle Squaring with Pieces of Small Boundary and Low Borel Complexity
• Classification of Local Problems on Paths from the Perspective of Descriptive Combinatorics
• Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms
• Borel asymptotic dimension and hyperfinite equivalence relations
• Measurable circle squaring
• Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem
• Scissor congruence

Discussions

• Alas, I have not been able to understand the papers on this subject well enough to answer your very reasonable questions. They can chop the disk and square into G_delta sets, which are countable inter [bsky, 0 points, 0 comments]

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• Borel Circle Squaring
• Circle Squaring with Pieces of Small Boundary and Low Borel Complexity
• Measurable circle squaring
• Measurable equidecompositions via combinatorics and group theory
• Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability