MEASURABLE REALIZATIONS OF ABSTRACT SYSTEMS OF CONGRUENCES
Skip to main content
eScholarship
Open Access Publications from the University of California

UCLA

UCLA Previously Published Works bannerUCLA

MEASURABLE REALIZATIONS OF ABSTRACT SYSTEMS OF CONGRUENCES

  • Author(s): Conley, Clinton T
  • Marks, Andrew S
  • Unger, Spencer T
  • et al.
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 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.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content
Current View