Sumset

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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.

[edit] 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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