Additive properties of subgroups of finite index in fields

In this paper, we confirm a conjecture of Bergelson and Shapiro concerning subgroups of finite index in multiplicative groups of fields which have maximal additive dimension. We also show that the natural extension of subgroups Gp of prime index p inside Q⁎ and additive dimension p+1 to the case where p is replaced by a composite integer n leads to subgroups of bounded additive dimension on a set of positive integers n of asymptotic density 1.