A famous result of Hausdorff states that a sphere with countably many points
removed can be partitioned into three pieces A,B,C such that A is congruent to
B (i.e., there is an isometry of the sphere which sends A to B), B is congruent
to C, and A is congruent to (B union C); this result was the precursor of the
Banach-Tarski paradox. Later, R. Robinson characterized the systems of
congruences like this which could be realized by partitions of the (entire)
sphere with rotations witnessing the congruences. The pieces involved were
nonmeasurable. In the present paper, we consider the problem of which systems
of congruences can be satisfied using open subsets of the sphere (or related
spaces); of course, these open sets cannot form a partition of the sphere, but
they can be required to cover "most of" the sphere in the sense that their
union is dense. Various versions of the problem arise, depending on whether one
uses all isometries of the sphere or restricts oneself to a free group of
rotations (the latter version generalizes to many other suitable spaces), or
whether one omits the requirement that the open sets have dense union, and so
on. While some cases of these problems are solved by simple geometrical
dissections, others involve complicated iterative constructions and/or results
from the theory of free groups. Many interesting questions remain open.Comment: 44 page