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An abstract system of congruences describes a way of partitioning a spaceinto finitely many pieces satisfying certain congruence relations. Examples ofabstract systems of congruences include paradoxical decompositions and$n$-divisibility of actions. We consider the general question of when there arerealizations of abstract systems of congruences satisfying variousmeasurability constraints. We completely characterize which abstract systems ofcongruences can be realized by nonmeager Baire measurable pieces of the sphereunder the action of rotations on the $2$-sphere. This answers a question ofWagon. We also construct Borel realizations of abstract systems of congruencesfor the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$.The combinatorial underpinnings of our proof are certain types of decompositionof Borel graphs into paths. We also use these decompositions to obtain someresults about measurable unfriendly colorings.

Measurable realizations of abstract systems of congruences