论文标题
圆与小边界和低虫的复杂性的圆形平方
Circle Squaring with Pieces of Small Boundary and Low Borel Complexity
论文作者
论文摘要
塔斯基（Tarski）从1925年开始的圆圈平方问题询问是否可以将飞机中的磁盘划分为有限的许多零件，并通过异构体重新组装它们，以产生同一区域正方形的分区。 1990年，Laczkovich最终在肯定中解决了它。最近，出现了一些新的证据，这些证据与结构化更好的零件相处：即可以衡量的碎片，并具有Baire（Grabowski-Máthé-Pikhurko）的特性，甚至是Borel（Marks-Unger）。 在本文中，我们表明，圆形的圆形片段是可能的圆形片段，其正边界的上限尺寸小于2（特别是，每块都是约旦可测量的）。我们还提高了碎片的骨质复杂性：即，我们表明每件可以被视为$f_σ$ sets的布尔组合。这是我们更普遍的结果的结果，该结果适用于$ r^k $，$ k \ ge 1 $的任何两个有限的子集，其边界的上限具有小于$ k $的上限。
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 $k$.
