Sumset

In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets ${\displaystyle A}$ and ${\displaystyle B}$ of an abelian group ${\displaystyle G}$ (written additively) is defined to be the set of all sums of an element from ${\displaystyle A}$ with an element from ${\displaystyle B}$. That is,

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

The ${\displaystyle n}$-fold iterated sumset of ${\displaystyle A}$ is

${\displaystyle nA=A+\cdots +A,}$

where there are ${\displaystyle n}$ 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

${\displaystyle 4\,\Box =\mathbb {N} ,}$

where ${\displaystyle \Box }$ 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 ${\displaystyle A+A}$ is small (compared to the size of ${\displaystyle A}$); 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.