Circle Squaring with Pieces of Small Boundary and Low Borel Complexity

TL;DR

This paper demonstrates that circle squaring can be achieved with Borel pieces that have positive Lebesgue measure and boundaries of Minkowski dimension less than 2, improving the structural complexity of the pieces.

Contribution

It introduces a method to partition a circle into Borel pieces with small boundary complexity, extending previous results to more regular pieces with lower Borel complexity.

Findings

Circle squaring with Borel pieces of positive measure

Boundaries have Minkowski dimension less than 2

Pieces are Boolean combinations of $F_σ$ sets

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 $k$.