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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On the maximum density of minimal asymptotic bases


Authors: Melvyn B. Nathanson and András Sárközy
Journal: Proc. Amer. Math. Soc. 105 (1989), 31-33
MSC: Primary 11B13; Secondary 11B05, 11P99
MathSciNet review: 973836
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Abstract: A set $ A$ of nonnegative integers is an asymptotic basis of order $ h$ if every sufficiently large integer is the sum of $ h$ elements of $ A$. It is proved that if $ A$ is an asymptotic basis of order $ h$ with lower asymptotic density $ {d_L}(A) > 1/h$, then there is a set $ W$ contained in $ A$ such that $ W$ has positive asymptotic density and $ A\backslash W$ is an asymptotic basis of order $ h$. This implies that if $ A$ is a minimal asymptotic basis of order $ h$, then $ {d_L}(A) \leq 1/h$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0973836-1
PII: S 0002-9939(1989)0973836-1
Keywords: Minimal asymptotic bases, additive bases, sumsets, additive number theory
Article copyright: © Copyright 1989 American Mathematical Society