$G_δ$ Circle Squaring
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arXiv
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| Format: | Preprint |
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2026
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| author | Unger, Spencer Varadarajan, Narmada Weilacher, Felix |
| author_facet | Unger, Spencer Varadarajan, Narmada Weilacher, Felix |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_19039 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $G_δ$ Circle Squaring Unger, Spencer Varadarajan, Narmada Weilacher, Felix Logic Combinatorics Metric Geometry 03E15, 52B45, 05C21 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. |
| title | $G_δ$ Circle Squaring |
| topic | Logic Combinatorics Metric Geometry 03E15, 52B45, 05C21 |
| url | https://arxiv.org/abs/2601.19039 |