$G_\delta$ Circle Squaring

Spencer Unger; Narmada Varadarajan; Felix Weilacher

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arXiv · 2601.19039

$G_\delta$ Circle Squaring

Spencer Unger·Narmada Varadarajan·Felix Weilacher

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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.

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BibTeX RIS

Spencer Unger, Narmada Varadarajan, Felix Weilacher. 2026-01-26. $G_\delta$ Circle Squaring. https://arxiv.org/abs/2601.19039

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