Sumset
# Sumset

In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets $A$ and $B$ of an abelian group $G$ (written additively) is defined to be the set of all sums of an element from $A$ with an element from $B$. That is,

    $A+B=\{a+b:a\in A,b\in B\}.$

The $n$-fold iterated sumset of $A$ is

    $nA=A+\cdots +A,$

where there are $n$ summands.

Many of the questions and results of additive combinatorics and additive number theory can be phrased in terms of sumsets. For example, Lagrange's four-square theorem can be written succinctly in the form

    $4\,\Box =\mathbb {N} ,$

where $\Box$ is the set of square numbers. A subject that has received a fair amount of study is that of sets with small doubling, where the size of the set $A+A$ is small (compared to the size of $A$); see for example Freiman's theorem.

