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,

The n-fold iterated sumset of A is

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

where ◻ 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.