Open sets satisfying systems of congruences

TL;DR

This paper investigates which systems of congruences can be satisfied by open subsets of spheres or similar spaces, exploring various conditions and using geometric and algebraic methods, with many questions still open for future research.

Contribution

It characterizes the realizability of systems of congruences with open sets on spheres, extending classical results and introducing new constructions and open problems.

Findings

Some systems are realizable by simple geometric dissections.

Complex iterative constructions are used for other systems.

Many open questions remain in the theory of open set congruences.

Abstract

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.