具有小边界和低Borel复杂度的圆方问题
Circle Squaring with Pieces of Small Boundary and Low Borel Complexity
AI总结：
本文提出使用具有小边界和低Borel复杂度的Borel块实现圆方问题。
摘要：
塔斯基于1925年提出的圆方问题询问是否可能将平面中的圆分成有限多块，并通过全等变换重新组装成面积相同的正方形。该问题最终在1990年由拉茨基维希肯定解决。最近，出现了一些新的证明，实现了结构更优的圆方：即具有勒贝格可测性且具有贝尔aire性质（Grabowski-Máthé-Pikhurko）或甚至为Borel集（Marks-Unger）的块。在本文中，我们证明可以使用具有正勒贝格测度且边界上Minkowski维数上界小于2（特别是每个块是雅各布可测的）的Borel块来实现圆方。我们还提高了块的Borel复杂度：即证明每个块可以取为F_σ集的布尔组合。这源于我们更一般的结果，适用于任何两个有界子集R^k（k≥1）且具有相同正测度且边界上Minkowski维数上界小于k的情况。
英文摘要：
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 $k$.
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