Let A and B be finite subsets of ℂ such that |B| = C|A|. We show the following variant of the sum product phenomenon: If |AB| < α|A| and α ≪ log |A|, then |k A + l B| ≫ |A|k|B|l. This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in . As an application of the case A = B we give a lower bound on |A+|+|A×|, where A+ is the set of sums of distinct elements of A and A× is the set of products of distinct elements of A.
© de Gruyter 2010