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| {{Short description|Set of pairwise sums of elements of two sets}}
| | #redirect [[Minkowski addition]] |
| {{inline |date=May 2024}}
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| In [[additive combinatorics]], the '''sumset''' (also called the [[Minkowski addition|Minkowski sum]]) of two [[subset]]s <math>A</math> and <math>B</math> of an [[abelian group]] <math>G</math> (written additively) is defined to be the set of all sums of an element from <math>A</math> with an element from <math>B</math>. That is,
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| :<math>A + B = \{a+b : a \in A, b \in B\}.</math>
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| The <math>n</math>-fold iterated sumset of <math>A</math> is
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| :<math>nA = A + \cdots + A,</math>
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| where there are <math>n</math> summands.
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| 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
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| :<math>4\,\Box = \mathbb{N},</math>
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| where <math>\Box</math> is the set of [[square number]]s. A subject that has received a fair amount of study is that of sets with ''small doubling'', where the size of the set <math>A+A</math> is small (compared to the size of <math>A</math>); see for example [[Freiman's theorem]].
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| ==See also==
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| *[[Restricted sumset]]
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| *[[Sidon set]]
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| *[[Sum-free set]]
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| *[[Schnirelmann density]]
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| *[[Shapley–Folkman lemma]]
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| *[[X + Y sorting]]
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| ==References==
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| *{{ 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
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| * {{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 }}
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| * {{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 }}
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| *Terence Tao and Van Vu, ''Additive Combinatorics'', Cambridge University Press 2006.
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| == External links ==
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| * {{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}}
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| [[Category:Sumsets| ]]
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| {{algebra-stub}}
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