Circle Squaring with Pieces of Small Boundary and Low Borel Complexity

TL;DR

This work proves circle squaring with a high degree of regularity by constructing a translation-based equidecomposition using Borel, Jordan-measurable pieces whose boundaries have upper Minkowski dimension strictly less than the ambient dimension. The authors reduce the problem to translations on the k‑torus, build a real flow from A to B, and then apply toast sequences to round the flow to an integral one, yielding a finite equidecomposition with controlled locality and complexity. The main contributions are (i) a boundary-dimension bound ζ ensuring Jordan measurability and (ii) a two‑level reduction of Borel complexity to Boolean combinations of Fσ sets, applicable to any bounded A,B with equal positive measure and sufficiently small boundary, with explicit bounds in the circle-square case. These results refine the understanding of the regularity of circle squarings and provide quantitative, describable descriptions of the equidecompositions, with potential implications for descriptive set theory and constructive decompositions.

Abstract

Tarski's Circle Squaring Problem from 1925 asks whether it is possible to partition a disk in the plane into finitely many pieces and reassemble them via isometries to yield a partition of a square of the same area. It was finally resolved by Laczkovich in 1990 in the affirmative. Recently, several new proofs have emerged which achieve circle squaring with better structured pieces: namely, pieces which are Lebesgue measurable and have the property of Baire (Grabowski-Máthé-Pikhurko) or even are Borel (Marks-Unger). In this paper, we show that circle squaring is possible with Borel pieces of positive Lebesgue measure whose boundaries have upper Minkowski dimension less than 2 (in particular, each piece is Jordan measurable). We also improve the Borel complexity of the pieces: namely, we show that each piece can be taken to be a Boolean combination of $F_σ$ sets. This is a consequence of our more general result that applies to any two bounded subsets of $R^k$, $k\ge 1$, of equal positive measure whose boundaries have upper Minkowski dimension smaller than $k$.

Figures (3)

• Figure 1: An illustration of the construction of the equidecomposition from an integer-valued flow $f$ in Lemma \ref{['outline:lem:matching']} in the simplified setting $d=2$. White nodes are elements of a maximally 19-discrete set and the boundaries of the induced Voronoi cells are in bold. Elements of $A$ and $B$ correspond to black round and square nodes, respectively. The total flow from the central Voronoi cell to the bottom-left cell is $2$; hence the first two elements of $A$ in the lexicographic order in the central Voronoi cell are associated to that cell. After distributing elements of $A$ and $B$ to the neighbouring cells, the two remaining (lexicographically latest) elements of $A$ and of $B$ in the central cell are mapped to one another.
• Figure 2: An example, for $d=2$, of a 4-cube $Q$, its dyadic partition $\mathcal{P}(Q)$ into $2$-subcubes and the flow $\phi_{\boldsymbol{u},Q}$ for a vertex $\boldsymbol{u}\in Q$ (where each arrow represents a flow of value $1/4$ in that direction).
• Figure 3: Two steps of the operation of rounding a flow $\phi$ along the boundary of a set $S$. The vertices within the grey region are in $S$. The edge $\boldsymbol{u}_s\boldsymbol{v}_s$ currently being rounded is depicted by a bold black line and the other two edges of the triangle containing $\boldsymbol{u}_s\boldsymbol{v}_s$ and $\boldsymbol{u}_{s+1}\boldsymbol{v}_{s+1}$ are depicted by a bold grey line. Some edges are labelled with numbers which represent their current flow value in the direction indicated by the arrow.

Theorems & Definitions (85)

• Theorem 1.1: Laczkovich Laczkovich92Laczkovich92b
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Laczkovich Laczkovich92b*Proof of Theorem 3; see also GrabowskiMathePikhurko17*Lemma 2.4
• Theorem 2.3: Integral Flow Theorem
• Remark 2.4
• Remark 2.5
• Definition 2.8
• Definition 2.9
• Lemma 2.11