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.

See also

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

References

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