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Sumset

In additive combinatorics, the sumset 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,



The n-fold iterated sumset of A is



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



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.

References

* Melvyn B. Nathanson, Additive Number Theory: Inverse Problems and Geometry of Sumsets volume 165 of GTM. Springer, 1996. Zbl 0859.11003.

* Terence Tao and Van Vu, Additive Combinatorics, Cambridge University Press 2006.

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