# 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,

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

The n-fold iterated sumset of A is

${\displaystyle nA=A+\cdots +A,}$

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

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

where ${\displaystyle \Box }$ 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.

## References

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• Terence Tao and Van Vu, Additive Combinatorics, Cambridge University Press 2006.