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,

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

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

:$4A\; =\; mathbb\{N\}$

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.