# On the Exact Order of Asymptotic Bases and Bases for Finite Cyclic Groups

• Xingde Jia
Chapter

## Summary

Let h be a positive integer, and A a set of nonnegative integers. A is called an exact asymptotic basis of order h if every sufficiently large positive integer can be written as a sum of h not necessarily distinct elements from A. The smallest such h is called the exact order of A, denoted by g(A). A subset AF of an asymptotic basis of order h may not be an asymptotic basis of any order. When AF is again an asymptotic basis, the exact order g(AF) may increase. Nathanson [48] studied how much larger the exact order g(AF) when finitely many elements are removed from an asymptotic basis of order h. Nathanson defines, for any given positive integers h and k,
$${G}_{k}(h) {=\max { }_{{ A \atop g(A)\leq h} }\max }_{F\in {I}_{k}(A)}g(A - F),$$
where $${I}_{k}(A) =\{ \vert F\vert \ \vert F\vert = k\text{ and }g(A - F) < \infty \}$$. Many results have been proved since Nathanson’s question was first asked in 1984. This function G k (h) is also closely related to interconnection network designs in network theory. This paper is a brief survey on this and few other related problems. G. Grekos [11] has a recent survey on a related problem.

## Keywords

Additive bases Asymptotic bases Exact asymptotic bases Extremal bases Finite cyclic groups Postage stamp problem

## Notes

### Acknowledgements

I like to thank Professor Mel Nathanson from whom I leaned combinatorial additive number theory, a wonderful and entertaining field of mathematics.

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