Abstract
The setA of nonnegative integers is a basis if every sufficiently large integerx can be written in the formx=a+a′ witha, a′∈A. IfA is not a basis, then it is a nonbasis. We construct a partition of the natural numbers into a basisA and a nonbasisB such that, as random elements are moved one at a time fromA toB, fromB toA, fromA toB, …, the setA oscillates from basis to nonbasis to basis … and the setB oscillates simultaneously from nonbasis to basis to nonbasis…
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Erdös, P., Nathanson, M.B. Partitions of the natural numbers into infinitely oscillating bases and nonbases. Commentarii Mathematici Helvetici 51, 171–182 (1976). https://doi.org/10.1007/BF02568149
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DOI: https://doi.org/10.1007/BF02568149