Simultaneous weighted sums of elements of finitely generated multiplicative groups

Let {G_j}_jεJ be a finite set of finitely generated subgroups of the multiplicative group of complex numbers C^x. Write H=∩ _jεJ G_j. Let n be a positive integer and a_ij a complex number for i = 1, ..., n and j ε J. Then there exists a set W with the following properties. The cardinality of W depends only on {G_j}_jεJ and n. If, for each jεJ, α has a representation α = Σ _iⁿ = ₁a _ijg_ij in elements g_ij of G_j, then α has a representation a= Σ_k=1ⁿ w_kh_k with w_kεW, h_k εH for k = 1,..., n. The theorem in this note gives information on such representations.