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

Abstract

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 A − F of an asymptotic basis of order h may not be an asymptotic basis of any order. When A − F is again an asymptotic basis, the exact order g(A − F) may increase. Nathanson [48] studied how much larger the exact order g(A − F) 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.