Set Addition and Set Multiplication

Let A be a subset of a ring with cardinality |A| = N . The sum set and the product set are 2A = A+A = {a1 +a2 | ai ∈ A} and A = AA = {a1a2 | ai ∈ A}. The study of sum-product sets is to estimate the sizes of 2A and A asymptotically when N is very large. Since Erdős and Szemerédi conjectured that if A ⊂ Z, at least one of |2A| and |A2| should be bigger than cN for any constant c, a lot of work has been done in this subject. In this paper, we give a survey of some of the main results in [C1]-[C5], and [BC1], [BC2]. In Section 1, we also put together the ideas in [C2]-[C4] and give a uniform treatment to some of the results there. Of particular importance is the role of Freiman’s Theorem and the Subspace Theorem. In Section 2, we consider the noncommutative analogue in matrix spaces. In Section 3, we report on results of sumproduct theorems for residue classes, especially prime fields. The results have striking applications to the theory of exponential sums, in particular Gauss sums of large degrees. In Section 4, we present a new and simpler proof of the Generalized Gauss sums Theorem for prime modulus [BGK], which is also included in the introduction of [BC2].