Concept

Sumset

Summary
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.
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