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# Sumset

In additive combinatorics, the sumset (also called the Minkowski sum) 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

\( 4\Box = \mathbb{N}, \)

where \( \Box \) 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.

See also

Minkowski addition (geometry)

Restricted sumset

Sidon set

Sum-free set

Schnirelmann density

Shapley–Folkman lemma

References

Henry Mann (1976). Addition Theorems: The Addition Theorems of Group Theory and Number Theory (Corrected reprint of 1965 Wiley ed.). Huntington, New York: Robert E. Krieger Publishing Company. ISBN 0-88275-418-1.

Nathanson, Melvyn B. (1990). "Best possible results on the density of sumsets". In Berndt, Bruce C.; Diamond, Harold G.; Halberstam, Heini et al. Analytic number theory. Proceedings of a conference in honor of Paul T. Bateman, held on April 25-27, 1989, at the University of Illinois, Urbana, IL (USA). Progress in Mathematics 85. Boston: Birkhäuser. pp. 395–403. ISBN 0-8176-3481-9. Zbl 0722.11007.

Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics 165. Springer-Verlag. ISBN 0-387-94655-1. Zbl 0859.11003.

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

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