Two classical results in Additive Combinatorics: Some related recent results

Let G be an abelian group, and A and B be finite subsets of G. The sumset A+B is the set of all elements of G that can be written in the form a+b, where $a \in A$ and 
$b \in B$. Given a subset A of G, determining properties of the h-fold sumset hA is a direct problem for addition in groups. In particular, Langrange's theorem that every nonnegative integer is a sum of four squares is an example of a direct problem. Also, for a finite set A, denoting its cardinality by |A|, finding a lower bound for |A+B| in terms of |A|and |B| is a direct problem. An inverse problem, on the other hand, is one where a knowledge of the size of hA gives some information about A.  In this lecture, first we have some introductory discussion about the nature of these problems. Then we take up a classical result in direct problems where G is the cyclic group of prime order and sketch a recent proof of it among other things. Next, as an application of this, we more on to discuss the  EGZ Theorem, a prototype of zerosum theorems. We shall also discuss on some related constants and some new developments.