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.