On additive bases

On Additive Bases

The critical number of G, denoted by c(G), is the smallest s such that Σ(S) = G for every subset S of G with cardinality s not containing 0. The parameter c(G) was first studied by Erdős and Heilbronn in [4]. They obtained the inequality c(Zp) ≤ 3 √ 6p. Olson proved in [13] that c(Zp) ≤ √ 4p− 3 + 1. The authors of [1] obtained the inequality c(Zp) ≤ √ 4p− 7. The evaluation of c(G) for groups wi...

On Additive Bases and Harmonious Graphs

This paper first considers several types of additive bases. A typical problem is to find nv(k), the largest n for which there exists a set {0 al < a2 <" < ak} Of distinct integers modulo n such that each in the range 0 =<-< n can be written at least once as mai + aj (modulo n) with </'. For example, nv(8) 24, The other problems arise if at least is changed to at most, or </' to-</', or if the w...

Essentialities in Additive Bases

Let A be an asymptotic basis for N0 of some order. By an essentiality of A one means a subset P such that A\P is no longer an asymptotic basis of any order and such that P is minimal among all subsets of A with this property. A finite essentiality of A is called an essential subset. In a recent paper, Deschamps and Farhi asked the following two questions : (i) does every asymptotic basis of N0 ...

Experimental Results on Additive 2-Bases

Additive Bases in Groups

We study the problem of removing an element from an additive basis in a general abelian group. We introduce analogues of the classical functions X,S and E (defined in the case of N) and obtain bounds on them. Our estimates on the functions SG and EG are valid for general abelian groups G while in the case of XG we show that distinct types of behaviours may occur depending on G.