Abstract
We construct an easily described family of partitions of the positive integers into n disjoint sets with essentially the same structure for every \(n \ge 2\). 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 in the second case we place some restrictions on the interaction but cannot obtain exhaustive results.
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Acknowledgements
The authors discovered these results while participating in an Illinois Geometry Lab group administered by Professors A.J. Hildebrand and K.B. Stolarsky. We thank them for advice on format and exposition. We also thank Professor Stolarsky for providing us with many of the references. We also thank the referee for comments that improved the readability of this paper.
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Krandel, J., Chen, W. Interpolating classical partitions of the set of positive integers. Ramanujan J 53, 209–241 (2020). https://doi.org/10.1007/s11139-019-00196-3
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DOI: https://doi.org/10.1007/s11139-019-00196-3