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MEASURABLE REALIZATIONS OF ABSTRACT SYSTEMS OF CONGRUENCES

Part of: Set theory Polytopes and polyhedra Ergodic theory Classical measure theory

Published online by Cambridge University Press:  24 February 2020

CLINTON T. CONLEY ,
ANDREW S. MARKS  and
SPENCER T. UNGER
CLINTON T. CONLEY
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA; clintonc@andrew.cmu.edu
ANDREW S. MARKS
Affiliation:
Department of Mathematics, University of California at Los Angeles, USA; marks@math.ucla.edu
SPENCER T. UNGER
Affiliation:
Department of Mathematics, Hebrew University of Jerusalem, Israel; unger.spencer@mail.huji.ac.il

Abstract

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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.

MSC classification

Primary: 03E15: Descriptive set theory
Secondary: 28A75: Length, area, volume, other geometric measure theory 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 52B45: Dissections and valuations (Hilbert's third problem, etc.)
Type
Foundations
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e10
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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