# 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.

## See also

- Minkowski addition (geometry)
- Restricted sumset
- Sidon set
- Sum-free set
- Schnirelmann density
- Shapley–Folkman lemma

## References

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