New Results on Subset Sums 3

In this paper we give a survey of recent results on sumsets of subsets of a field with polynomial restrictions. Let F be a field and let F× be the multiplicative group F \ {0}. The additive order of the (multiplicative) identity of F is either infinite or a prime, we call it the characteristic of F . Let A and B be finite subsets of the field F . Set A+B = {a+ b: a ∈ A and b ∈ B} and AuB = {a+ b: a ∈ A, b ∈ B, and a 6= b}. The theorem of Cauchy and Davenport asserts that if F is the field of residues modulo a prime p, then |A+B| ≥ min{p, |A|+ |B| − 1}. In 1964 Erdős and Heilbronn conjectured that in this case |AuA| ≥ min{p, 2|A| − 3}, 1