Sumset

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In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets ${\displaystyle A}$ and ${\displaystyle B}$ of an abelian group ${\displaystyle G}$ (written additively) is defined to be the set of all sums of an element from ${\displaystyle A}$ with an element from ${\displaystyle B}$. That is,

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

The ${\displaystyle n}$-fold iterated sumset of ${\displaystyle A}$ is

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

where there are ${\displaystyle n}$ 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 ${\displaystyle A+A}$ is small (compared to the size of ${\displaystyle A}$); see for example Freiman's theorem.