sumset

{{Short description|Set of pairwise sums of elements of two sets}}

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.

References

{{ cite book | author=Henry Mann | authorlink=Henry Mann | title=Addition Theorems: The Addition Theorems of Group Theory and Number Theory | publisher=Robert E. Krieger Publishing Company | url=http://www.krieger-publishing.com/subcats/MathematicsandStatistics/mathematicsandstatistics.html | location=Huntington, New York | year=1976 | edition=Corrected reprint of 1965 Wiley | isbn=0-88275-418-1

}}

{{cite book | zbl=0722.11007 | last=Nathanson | first=Melvyn B. | chapter=Best possible results on the density of sumsets | pages=395–403 | editor1-last=Berndt | editor1-first=Bruce C. | editor1-link=Bruce C. Berndt | editor2-last=Diamond | editor2-first=Harold G. | editor3-last=Halberstam | editor3-first=Heini | editor3-link=Heini Halberstam |display-editors = 3 | editor4-last=Hildebrand | editor4-first=Adolf | title=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) | series=Progress in Mathematics | volume=85 | location=Boston | publisher=Birkhäuser | year=1990 | isbn=0-8176-3481-9 }}
{{cite book | first=Melvyn B. | last=Nathanson | title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets | volume=165 | series=Graduate Texts in Mathematics | publisher=Springer-Verlag | year=1996 | isbn=0-387-94655-1 | zbl=0859.11003 }}
Terence Tao and Van Vu, Additive Combinatorics, Cambridge University Press 2006.

External links

{{Cite web |last=Sloman |first=Leila |date=2022-12-06 |title=From Systems in Motion, Infinite Patterns Appear |url=https://www.quantamagazine.org/infinite-patterns-appear-in-numbers-described-as-moving-systems-20221205/ |website=Quanta Magazine |language=en}}

sumset

See also

References

External links