• Source: Sumset

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



A


{\displaystyle A}

and



B


{\displaystyle B}

of an abelian group



G


{\displaystyle G}

(written additively) is defined to be the set of all sums of an element from



A


{\displaystyle A}

with an element from



B


{\displaystyle B}

. That is,




A
+
B
=
{
a
+
b
:
a
∈
A
,
b
∈
B
}
.


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


The



n


{\displaystyle n}

-fold iterated sumset of



A


{\displaystyle A}

is




n
A
=
A
+
⋯
+
A
,


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


where there are



n


{\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




4

◻
=

N

,


{\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



A
+
A


{\displaystyle A+A}

is small (compared to the size of



A


{\displaystyle A}

); see for example Freiman's theorem.




See also






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.

Kata Kunci Pencarian:

• Daftar masalah matematika yang belum terpecahkan
• Sumset
• Restricted sumset
• Additive combinatorics
• Erdős sumset conjecture
• Additive number theory
• Freiman's theorem
• List of conjectures by Paul Erdős
• Arithmetic combinatorics
• Sum-free set
• Erdős–Szemerédi theorem