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Measurable circle squaring. (English) Zbl 1376.52023

Two sets \(A,B\subseteq\mathbb{R}^k\) are called equidecomposable if there are a partition \(A=A_1\cup\cdots\cup A_n\) (into finitely many parts) and isometries \(\gamma_1,\dots,\gamma_n\) of \(\mathbb{R}^k\) such that the images of the parts \(\gamma_1(A_1),\dots,\gamma_n(A_n)\) partition \(B\). The sets \(A,B\subseteq\mathbb{R}^k\) are called equivalent if they are equidecomposable using translations.
A question by Tarski in 1925 asks if the disk and the square in \(\mathbb{R}^2\) of the same area are equidecomposable. Different answers, by assuming different properties of the parts or the used transformations have been given (see the introduction of the paper under review for an extensive account on this). The authors broaden these answers by proving that the square and the disc (with the same area) are equivalent and the sets of the partition are Baire and Lebesgue measurable.
The result follows from the main result of the paper, which states that for \(k\geq 1\), if \(A,B\subset\mathbb{R}^k\) are bounded sets with nonempty interior with the same Lebesgue measure, \(\dim_{\square}(\partial A)<k\) and \(\dim_{\square}(\partial B)<k\), then \(A\) and \(B\) are equivalent with parts that are both Baire and Lebesgue measurable. Here \(\dim_{\square}(X)\) denotes the box (or grid or upper Minkowski) dimension of \(X\subset\mathbb{R}^n\). The question if the parts can be taken as Borel sets has been recently proved by A. S. Marks and S. T. Unger [ibid. 186, No. 2, 581–605 (2017; Zbl 1400.03064)].
To prove the result, the authors first reduce the problem to the torus and use there some results of M. Laczkovich [Rend. Ist. Mat. Univ. Trieste 23, 145–176 (1991; Zbl 0801.28001)]. Laczkovich has proved that the answer to Tarski question is affirmative and that the square and circle of the same area are equivalent but the pieces need not be measurable. Infinite bipartite graphs are also used.

MSC:

52B45 Dissections and valuations (Hilbert’s third problem, etc.)
28A75 Length, area, volume, other geometric measure theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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