Partitioning an Interval Into Finitely Many Congruent Parts

Abstract

A subset of dense additive subgroup of real numbers will be called an interval if it is nonnull and bounded and is convex with respect to the given subgroup. By a partition of an interval we shall mean a proper decomposition of it into finitely many subsets, called parts, which are disjoint and congruent by pairs. We allow congruence by reflection as well as by translation in superposing the parts of a partition. With regard to partitioning two types of interval are to be distinguished: pivotal and nonpivotal. An interval will be called pivotal if it is symmetric but does not contain its midpoint; otherwise it will be called nonpivotal. The purpose of this paper is to prove the following two propositions concerning partitions of a nonpivotal interval: (1) Every part of a partition of a nonpivotal interval is merely a finite union of subintervals. (2) If a nonpivotal interval can be partitioned into n parts, it can also be partitioned into n subintervals. Let us see how these results apply to partitioning an interval in the group of all real numbers. Such an interval is clearly nonpivotal; it may, however, be either symmetric or asymmetric. If the interval is asymmetric, it admits many partitions, but according to (1) the parts of any partition of it consist of finite unions of subintervals hence are measurable. On the other hand, if the interval is symmetric, it evidently cannot be partitioned into subintervals and so by (2) cannot be partitioned at all. We mention in contrast to this that, as von Neumann2 has shown, every nondegenerate real interval, whether symmetric or asymmetric, can be decomposed by translation into denumerably many congruent necessarily nonmeasurable sets.