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 $n$ -divisibility of actions. 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. This answers a question by Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_{2}(\mathbb{Z})$ on $\mathsf{P}^{1}(\mathbb{R})$ . The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.

中文翻译：

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一致的抽象系统的可测量的实现

一个抽象的全等系统描述了一种将空间划分为满足某些全等关系的有限多块的方法。同余抽象系统的例子包括悖论分解和 $n$ - 行动的可分性。我们考虑何时实现满足各种可测量性约束的同余抽象系统的一般问题。我们完全描述了哪些抽象的同余系统可以通过球体的非微薄 Baire 可测量部分在旋转的作用下实现 $2$ -领域。这回答了 Wagon 的一个问题。我们还构建了同余抽象系统的 Borel 实现，用于 $\mathsf{PSL}_{2}(\mathbb{Z})$ 在 $\mathsf{P}^{1}(\mathbb{R})$ . 我们证明的组合基础是将某些类型的 Borel 图分解为路径。我们还使用这些分解来获得一些关于可测量的不友好着色的结果。