Interpolating Classical Partitions of the Set of Positive Integers

We construct an easily described family of partitions of the positive integers into $n$ disjoint sets with essentially the same structure for every $n \geq 2$. In a special case, it interpolates between the Beatty $\frac{1}{\phi} + \frac{1}{\phi^2} = 1$ partitioning ($n=2$) and the 2-adic partitioning in the limit as $n \rightarrow \infty$. We then analyze how membership of elements in the sets of one partition relates to membership in the sets of another. We investigate in detail the interactions of two Beatty partitions with one another and the interactions of the $\phi$ Beatty partition mentioned above with its "extension" to three sets given by the construction detailed in the first part. In the first case, we obtain detailed results whereas the second case we place some restrictions on the interaction but cannot obtain exhaustive results.