In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets and of an abelian group (written additively) is defined to be the set of all sums of an element from with an element from . That is,
The -fold iterated sumset of is
where there are 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
where 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 is small (compared to the size of ); see for example Freiman's theorem.
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