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# Additive Systems and a Theorem of de Bruijn

Melvyn B. Nathanson
The American Mathematical Monthly
Vol. 121, No. 1 (January 2014), pp. 5-17
DOI: 10.4169/amer.math.monthly.121.01.005
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.121.01.005
Page Count: 13
Item Type
Article
References
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## Abstract

Abstract This paper proves a theorem of de Bruijn that classifies additive systems for the nonnegative integers, that is, families \documentclass{article} \pagestyle{empty}\begin{document} $\mathcal{A} = (A_i)_{i\in I}$ \end{document} of sets of nonnegative integers, each set containing 0, such that every nonnegative integer can be written uniquely in the form \documentclass{article} \pagestyle{empty}\begin{document} $\sum_{i\in I} a_i$ \end{document} , with \documentclass{article} \pagestyle{empty}\begin{document} $a_i \in A_i$ \end{document} for all i, and \documentclass{article} \pagestyle{empty}\begin{document} $a_i \neq 0$ \end{document} for only finitely many i.