You need to know: Euclidean plane , notation for the translation of set by a vector , open subset of , countable union, countable intersection, and set difference () of sets. You also need to know what is the Axiom of choice to fully understand the context.
Background: A subset is a Borel set if it can be formed from open sets through the operations of countable union, countable intersection, and set difference. We call two sets equidecomposable by translations if there are partitions and , such that , , for some vectors . If, moreover, all (and thus ) are Borel sets, we say that A and B are equidecomposable by translations with Borel parts.
The Theorem: On 17th December 2016, Andrew Marks and Spencer Unger submitted to arxiv a paper in which they proved that a circle and a square of the same area on the plane are equidecomposable by translations with Borel parts.
Short context: In 1990, Laczkovich, answering a 1925 question of Tarski, proved that circle and a square of the same area are equidecomposable by translations. In a paper submitted in 2015, Grabowski, Máthé, and Pikhurko proved that this is possible even using only Lebesgue measurable pieces (that is, those having a well-define area). However, both results use axiom of choice and the resulting pieces are impossible to construct explicitly. The Theorem states that the circle can be squared with only Borel pieces. The proof does not use the axiom of choice. If such a proof is possible, we say that a problem has a constructive solution.
Links: Free arxiv version of the original paper is here, journal version is here.
Theorems of the 21st century 