Solutions to congruences using sets with the property of Baire

TL;DR

This paper characterizes systems of congruences realizable by partitions of spheres or similar spaces into sets with the property of Baire, extending results related to the Banach-Tarski paradox and using similar methods.

Contribution

It provides a characterization of congruence systems solvable with Baire property sets under broader isometry groups, extending previous work on the Banach-Tarski paradox.

Findings

Characterization of systems of congruences realizable with Baire property sets.

Extension of Banach-Tarski paradox results to broader isometry groups.

Use of similar methods as previous results to achieve these characterizations.

Abstract

Hausdorff's paradoxical decomposition of a sphere with countably many points removed (the main precursor of the Banach-Tarski paradox) actually produced a partition of this set into three pieces A,B,C such that A is congruent to B (i.e., there is an isometry of the set which sends A to B), B is congruent to C, and A is congruent to (B union C). While refining the Banach-Tarski paradox, R. Robinson characterized the systems of congruences like this which could be realized by partitions of the sphere with rotations witnessing the congruences. The purpose of the present paper is to characterize those systems of congruences which can be satisfied by partitions of the sphere or related spaces (any complete metric space acted on in a sufficiently free way by a free group of homeomorphisms) into sets with the property of Baire. Dougherty and Foreman proved that the Banach-Tarski paradox can be achieved using such sets, and gave versions of this result using open sets and related results about partitions of spaces into congruent sets. The same methods are used here. We also characterize the systems solvable on the sphere using sets with the property of Baire but allowing all isometries (instead of just rotations).