Editing Sumset (section)

{{Short description|Set of pairwise sums of elements of two sets}}
{{inline |date=May 2024}}
In [[additive combinatorics]], the '''sumset''' (also called the [[Minkowski addition|Minkowski sum]]) of two [[subset]]s <math>A</math> and <math>B</math> of an [[abelian group]] <math>G</math> (written additively) is defined to be the set of all sums of an element from <math>A</math> with an element from <math>B</math>. That is,

:<math>A + B = \{a+b : a \in A, b \in B\}.</math>

The <math>n</math>-fold iterated sumset of <math>A</math> is

:<math>nA = A + \cdots + A,</math>

where there are <math>n</math> 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

:<math>4\,\Box = \mathbb{N},</math>

where <math>\Box</math> is the set of [[square number]]s. A subject that has received a fair amount of study is that of sets with ''small doubling'', where the size of the set <math>A+A</math> is small (compared to the size of <math>A</math>); see for example [[Freiman's theorem]].