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A NOTE ON THE ERDŐS–GRAHAM THEOREM

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

Abstract

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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References

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[1] Chen, S. and Gu, W. Z., ‘Exact order of subsets of asymptotic bases’, J. Number Theory 41 (1992), 1521.
[2] Erdős, P. and Graham, R. L., ‘On bases with an exact order’, Acta Arith. 37 (1980), 201207.
[3] Jia, X. D., ‘Exact order of subsets of asympotic bases in additive number theory’, J. Number Theory 28 (1988), 205218.
[4] Nash, J. C. M. and Nathanson, M. B., ‘Cofinite subsets of asymptotic bases for the positive integers’, J. Number Theory 20 (1985), 363372.
[5] Nathanson, M. B., ‘The exact order of subsets of additive bases’, in: Number Theory (New York, 1982), Lecture Notes in Mathematics, 1052 (Springer, Berlin, 1984), 273277.
[6] Plagne, A., ‘Removing one element from an exact additive basis’, J. Number Theory 87 (2001), 306314.
[7] Yang, Q. H. and Chen, F. J., ‘On bases with a T-order’, Integers 11 (2011), A5.
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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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