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 %2B B = \{a%2Bb�: a \in A, b \in B\}.

The n-fold iterated sumset of A is

nA = A %2B \cdots %2B 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

4A = \mathbb{N}

where A 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

References