Circle Squaring with Pieces of Small Boundary and Low Borel Complexity

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-M\'ath\'e-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 $F_\sigma$ 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 $k$.