# Measurable realizations of abstract systems of congruences

## TL;DR

This paper characterizes when abstract systems of congruences can be realized with measurable pieces under group actions, providing solutions to open questions and constructing specific Borel realizations.

## Contribution

It offers a complete characterization for realizations on the 2-sphere and constructs Borel realizations for the action of PSL(2,Z), advancing understanding of measurable congruence systems.

## Key findings

- Characterization of realizability on the 2-sphere with nonmeager Baire measurable pieces.
- Construction of Borel realizations for PSL(2,Z) acting on the projective line.
- Development of graph decomposition techniques for measurable colorings.

## 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 $n$-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 of 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.