Sumset

Template:Short description Template:Inline In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets <math>A</math> and <math>B</math> of an abelian group <math>G</math> (written additively) is defined to be the set of all sums of an element from <math>A</math> with an element from <math>B</math>. That is,

<math>A + B = \{a+b : a \in A, b \in B\}.</math>

The <math>n</math>-fold iterated sumset of <math>A</math> is

<math>nA = A + \cdots + A,</math>

where there are <math>n</math> 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

<math>4\,\Box = \mathbb{N},</math>

where <math>\Box</math> 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 <math>A+A</math> is small (compared to the size of <math>A</math>); see for example Freiman's theorem.

See also

• Restricted sumset
• Sidon set
• Sum-free set
• Schnirelmann density
• Shapley–Folkman lemma
• X + Y sorting

References

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

External links

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