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Borel circle squaring. (English) Zbl 1400.03064
This review is taken from the the authors’ introduction (with some omissions and minor adaptations): the reviewer considers it as most informative of the content and the aim of the paper. He would like to express his appreciation for their remarkable result.
In 1925, Tarski posed the problem of whether a disk and a square of the same area in the plane are equidecomposable by isometries. This problem became known as Tarski’s circle problem. In contrast to the Banach-Tarski paradox in $$\mathbb R^3$$, a theorem of Tarski implies that any two Lebesgue measurable sets in $$\mathbb R^2$$ that are equidecomposable by isometries must have the same Lebesgue measure, even when the pieces used in the equidecomposition are allowed to be nonmeasurable. Thus the requirement that the circle and the square have the same area is necessary.
The well-known Wallace-Bolyai-Gerwien theorem states that two polygons in $$\mathbb R^2$$ have the same area if and only if they dissection congruent, that is, equidecomposable by polygonal pieces where we may ignore boundaries. Hilbert’s third problem asked whether any two polyedra of the same volume are dissection congruent. Dehn famously gave a negative answer to this problem.
M. Laczkovich answered Tarski’s question positively in 1990 [J. Reine Angew. Math. 404, 77–117 (1990; Zbl 0748.51017)], using only translates in his equidecomposition. In [J. London Math. Soc. 46, No. 1, 58–64 (1992; Zbl 0776.11041)], he improved this result to give a very general sufficient condition for when two bounded sets in $$\mathbb R^k$$ of the same Lebesgue measure are equidecomposable by translations.
Theorem [Laczkovich, 1992, loc. cit.]. If $$k \geq 1$$ and $$A,B\subseteq \mathbb R^k$$ are bounded sets with the same positive Lebesgue measure whose boudaries have upper Minkowski dimension less than $$k$$, then $$A$$ and $$B$$ are equidecomposable by translations.
Laczkovich’s proofs are nonconstructive and use the axiom of choice. It remained an open problem whether such equidecompositions could be done constructively. Wagon made this question precise by asking whether Tarski’s circle squaring problem could be solved using Borel pieces.
In this paper, the authors answer Wagon’s question and give a completely constructive solution to Tarski’s circle squaring problem. More generally, they prove a Borel version of Laczkovich’s theorem, thus providing also a “Borel solution” to Hilbert’s third problem: any two bounded sets in $$\mathbb R^k$$ with “small boundary” have the same measure if and only if they are equidecomposable by translations using Borel pieces:
Theorem. If $$k\geq 1$$ and $$A,B\subseteq \mathbb R^k$$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $$k$$, then $$A$$ and $$B$$ are equidecomposable by translations using Borel pieces.
A key idea in their proof is to use flows in infinite graphs as an intermediate step towards constructing equidecompositions. Laczkovich’s discrepancy estimates – the central ingredient in the proof of Laczkovich’s theorem – are used to show the convergence of this construction.
Another important tool in the proof comes from the theory of orbit equivalence and Borel equivalence relations. In particular, it is used a result of S. Gao et al. [“Forcing constructions and countable Borel equivalence relations”, Preprint, arXiv:1503.07822] about special types of witnesses to the hyperfiniteness of free Borel actions on $$\mathbb Z^d$$: this hyperfiniteness witness is used to turn their real-valued Borel flow between $$A$$ and $$B$$ into an integer-valued flow. This step in the proof also relies on the integral flow theorem, which is a corollary of the Ford-Fulkerson proof of the max-flow min-cut theorem.

##### MSC:
 03E15 Descriptive set theory 05C21 Flows in graphs 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 52B45 Dissections and valuations (Hilbert’s third problem, etc.)
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