Complex Circles of Partition and the Expansion Principles

In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to partitions defined within the complex plane, which now serves as both the base and bearing set. We employ the squeeze principle as a central tool for rigorously investigating the possibility to partition numbers with base set as a certain subset of the complex plane.