A constructive solution to Tarski’s circle squaring problem

Abstract: In 1925, Tarski asked whether a disk in R^2 can be partitioned into finitely many pieces which can be rearranged by isometries to form a square of the same area.  The restriction of having a disk and a square with the same area is necessary.  In 1990, Laczkovich gave a positive answer to the problem using the axiom of choice.  We give a completely explicit (Borel) way to break the circleand the square into congruent pieces.  This answers a question of Wagon.  Our proof has three main components.  The first is work of Laczkovich in Diophantine approximation.  The second is recent progress in a program of descriptive set theory to understand the complexity of actions of amenable groups.  The third is the study of flows in networks.