Abstract
In 1925, Tarski posed the problem of whether a disc in R2 can be
partitioned into finitely many pieces which can be rearranged by
isometries to form a square of the same area. Unlike the Banach-Tarski
paradox in R3, it can be shown that two Lebesgue measurable sets in
R2 cannot be equidecomposed by isometries unless they have the same
measure. Hence, the disk and square must necessarily be of the same
area.
In 1990, Laczkovich showed that Tarski's circle squaring problem has a
positive answer using the axiom of choice. We give a completely
constructive solution to the problem and describe an explicit (Borel)
way to equidecompose a circle and a square. This answers a question of
Wagon.
Our proof has three main ingredients. The first is work of Laczkovich
in Diophantine approximation. The second is recent progress in a
research program in descriptive set theory to understand how the
complexity of a countable group is related to the complexity of the
equivalence relations generated by its Borel actions. The third
ingredient is ideas coming from the study of flows in networks.
This is joint work with Spencer Unger.