Erdos and Szemeredi conjectured that, for any finite set A of integers, either the set of sums of pairs of elements of A is nearly as large as possible, or the set of products of pairs of elements of A is nearly as large as possible. I will discuss what is known for this problem when A is a set of real numbers, complex numbers or quaternions, along with some techniques that have been useful for this question. I will mention recent joint work with Abdul Basit, which gives the best known result when A is a set of complex numbers or quaternions.
Ben Lund