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A NOTE ON ASYMPTOTIC NONBASES

Part of: Sequences and sets

Published online by Cambridge University Press:  13 October 2016

DENG-RONG LING
DENG-RONG LING*
Affiliation:
School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, PR China email [email protected]
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Abstract

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Let $A$ be a subset of $\mathbb{N}$ , the set of all nonnegative integers. For an integer $h\geq 2$ , let $hA$ be the set of all sums of $h$ elements of $A$ . The set $A$ is called an asymptotic basis of order $h$ if $hA$ contains all sufficiently large integers. Otherwise, $A$ is called an asymptotic nonbasis of order $h$ . An asymptotic nonbasis $A$ of order $h$ is called a maximal asymptotic nonbasis of order $h$ if $A\cup \{a\}$ is an asymptotic basis of order $h$ for every $a\notin A$ . In this paper, we construct a sequence of asymptotic nonbases of order $h$ for each $h\geq 2$ , each of which is not a subset of a maximal asymptotic nonbasis of order $h$ .

Keywords

asymptotic nonbasismaximal asymptotic nonbasis

MSC classification

Primary: 11B13: Additive bases, including sumsets
Secondary: 11B34: Representation functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 1 - 4
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

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