Skip to main content Accessibility help
×
Home

A NOTE ON ASYMPTOTIC NONBASES

  • DENG-RONG LING (a1)

Abstract

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 $h$ .

Copyright

References

Hide All
[1] Alladi, K. and Krantz, S., ‘Reflections on Paul Erdős on his birth centenary’, Notices Amer. Math. Soc. 62 (2015), 121143.
[2] Erdős, P. and Nathanson, M. B., ‘Maximal asymptotic nonbases’, Proc. Amer. Math. Soc. 48 (1975), 5760.
[3] Erdős, P. and Nathanson, M. B., ‘Oscillations of bases for the natural numbers’, Proc. Amer. Math. Soc. 53 (1975), 253258.
[4] Hennefeld, J., ‘Asymptotic nonbases which are not subsets of maximal asymptotic nonbases’, Proc. Amer. Math. Soc. 62 (1977), 2324.
[5] Nathanson, M. B., ‘Minimal bases and maximal nonbases in additive number theory’, J. Number Theory 6 (1974), 324333.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed