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Limits and Decomposition of de Bruijn’s Additive Systems

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 220)

Abstract

An additive system for the nonnegative integers is a family \((A_i)_{i\in I}\) of sets of nonnegative integers with \(0 \in A_i\) for all \(i \in I\) such that every nonnegative integer can be written uniquely in the form \(\sum _{i\in I} a_i\) with \(a_i \in A_i\) for all i and \(a_i \ne 0\) for only finitely many i. In 1956, de Bruijn proved that every additive system is constructed from an infinite sequence \((g_i )_{i \in \mathbf N}\) of integers with \(g_i \ge 2\) for all i or is a contraction of such a system. This paper discusses limits and the stability of additive systems and also describes the “uncontractable” or “indecomposable” additive systems.

Keywords

  • Additive system
  • Additive basis
  • British number system

2010 Mathematics Subject Classification:

  • 11A05
  • 11B75

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References

  1. N.G. de Bruijn, On number systems. Nieuw Arch. Wisk. 4(3), 15–17 (1956)

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  2. M. Maltenfort, Characterizing additive systems. Am. Math. Monthly 124, 132–148 (2017)

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  3. A.O. Munagi, k-complementing subsets of nonnegative integers. Int. J. Math. Math. Sci. 215–224 (2005)

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  4. M.B. Nathanson, Additive systems and a theorem of de Bruijn. arXiv:1301.6208 (2013)

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Acknowledgements

Supported in part by a grant from the PSC-CUNY Research Award Program.

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Correspondence to Melvyn B. Nathanson .

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Nathanson, M.B. (2017). Limits and Decomposition of de Bruijn’s Additive Systems. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_18

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