On additive bases III
Volume 193 / 2020
Abstract
Let be an additive finite abelian group and S a sequence over G. We say that S is regular if for every proper subgroup H\subset G, S contains at most |H|-1 terms from H. Let \mathsf c_0(G) be the smallest integer t such that every regular sequence S over G of length |S|\ge t forms an additive basis of G, i.e., every element of G can be expressed as a sum over a nonempty subsequence of S. Some related problems on \mathsf c_0(G) have been studied by Mann and Olson around 1968. Answering a problem proposed by Olson, Peng determined \mathsf c_0(G) for all elementary finite abelian groups in 1987. Recently, several authors have studied \mathsf c_0(G) for general abelian groups, and so far the invariant \mathsf c_0(G) has been determined for some finite abelian groups including the cyclic groups, the groups of even order, the groups of rank at least five, the groups of rank in \{3, 4\} and with the smallest prime divisor of |G| at least 11, and the p-groups of rank 2. In this paper, we shall determine \mathsf c_0(G) for all groups such that the rank of G is in \{3, 4\}, the smallest prime divisor of |G| does not exceed 7 and |G|\ge 3.72\times 10^7.
