Simultaneous weighted sums of elements of finitely generated multiplicative groups

Abstract

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 i n = 1a ijg ij in elements g ij of G j , then α has a representation a= Σ k=1 n w kh k with w kεW, h k εH for k = 1,..., n. The theorem in this note gives information on such representations.