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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Oscillations of bases for the natural numbers


Authors: Paul Erdős and Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 53 (1975), 253-258
MSC: Primary 10L05
DOI: https://doi.org/10.1090/S0002-9939-1975-0384739-8
MathSciNet review: 0384739
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Abstract: Let $ A$ be a set of positive integers. Then $ A$ is a basis if every sufficiently large integer $ n$ can be written in the form $ n = {a_i} + {a_j}$ with $ {a_i},\;{a_j}\epsilon A$. Otherwise, $ A$ is a nonbasis. In this paper we construct sets which oscillate from basis to nonbasis to basis or from nonbasis to basis to nonbasis under finite perturbations of the sets.


References [Enhancements On Off] (What's this?)

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  • [2] E. Hartter, Ein Beitrag zur Theorie der Minimalbasen, J. Reine Angew. Math. 196 (1956), 170-204. MR 19, 122. MR 0086086 (19:122a)
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  • [4] A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe. I, II, J. Reine Angew. Math. 194 (1955), 40-65, 111-140. MR 17, 713. MR 0075228 (17:713a)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0384739-8
Keywords: Minimal basis, maximal nonbasis, sumsets of integers, oscillations of bases, additive number theory
Article copyright: © Copyright 1975 American Mathematical Society

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