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

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

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. (eds.). 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. Vol. 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. Vol. 165. Springer-Verlag. ISBN 0-387-94655-1. Zbl 0859.11003.
• Terence Tao and Van Vu, Additive Combinatorics, Cambridge University Press 2006.

External links

• Sloman, Leila (2022-12-06). "From Systems in Motion, Infinite Patterns Appear". Quanta Magazine.