Abstract
An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and partitioning a set into n congruent pieces. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the 2-sphere. We also construct Borel realizations of abstract systems of congruences for the action of PSL₂(Z) on the projective line. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths.