**S. D. Adhikari**

Harish-Chandra Research Institute, Allahabad

April 7, 2011

**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, Lagrange'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 move on to