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Proceedings of the American Mathematical Society

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

 

 

Maximal asymptotic nonbases


Authors: Paul Erdős and Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 48 (1975), 57-60
DOI: https://doi.org/10.1090/S0002-9939-1975-0357363-0
MathSciNet review: 0357363
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Abstract: Let $ A$ be a set of nonnegative integers. If all but a finite number of positive integers can be written as a sum of $ h$ elements of $ A$, then $ A$ is an asymptotic basis of order $ h$. Otherwise, $ A$ is an asymptotic nonbasis of order $ h$. A class of maximal asymptotic nonbases is constructed, and it is proved that any asymptotic nonbasis of order 2 that satisfies a certain finiteness condition is a subset of a maximal asymptotic nonbasis of order 2.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0357363-0
Keywords: Addition of sequences, sum sets, asymptotic bases, asymptotic nonbases, maximal nonbases
Article copyright: © Copyright 1975 American Mathematical Society