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  • WENHUI WANG (a1) and MIN TANG (a2)


Let ${\mathcal{A}}=\{a_{1}<a_{2}<\cdots \,\}$ be a set of nonnegative integers. Put $D({\mathcal{A}})=\gcd \{a_{k+1}-a_{k}:k=1,2,\ldots \}$ . The set ${\mathcal{A}}$ is an asymptotic basis if there exists $h$ such that every sufficiently large integer is a sum of at most $h$ (not necessarily distinct) elements of ${\mathcal{A}}$ . We prove that if the difference of consecutive integers of ${\mathcal{A}}$ is bounded, then ${\mathcal{A}}$ is an asymptotic basis if and only if there exists an integer $a\in {\mathcal{A}}$ such that $(a,D({\mathcal{A}}))=1$ .


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This work was supported by the National Natural Science Foundation of China (Grant No.11471017).



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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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