Paper detail

Solutions to congruences using sets with the property of Baire

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).