en>Ipigott: /* References */ cat

2014-10-04T07:30:12Z

References: cat

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	In [[additive combinatorics]], the '''sumset''' (also called the [[Minkowski addition\|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,

	~~:<math>A + B = \{a+b : a \in A, b \in B\}.</math>~~

	~~The '''''n''-fold iterated sumset''' of ''A'' is~~		[http://www.sharkbayte.com/keyword/Love+Seibert Love Seibert] is the name my parents gave me and I adore it. Hawaii is the only place I've been residing in. I am truly fond of bottle tops collecting and now I'm attempting to make cash with it. [http://www.thefreedictionary.com/Dispatching Dispatching] is how he supports his family members. I've been working on my website for some time now. Check it out here: http://www.bankami.net/2013/12/ban-bencana-banjir-di-kuantan-disember.html<br><br>Also visit my homepage: Nya online casinon 2014 [[http://www.bankami.net/2013/12/ban-bencana-banjir-di-kuantan-disember.html klikk gjennom følgende side]]

	:~~<math>nA = A + \cdots + A,<~~/~~math>~~

	~~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~~

	~~:<math>4\Box = \mathbb{N},<~~/~~math>~~

	~~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~~ '~~'A'' + ''A'' is small (compared~~ to ~~the size of ''A''); see for example [[Freiman's theorem]]~~.

	~~==See also==~~
	*[[Minkowski addition]] ([~~[geometry]])~~
	*[[Restricted sumset]]
	*[[Sidon set]]
	*[[Sum-free set]]
	*[[Schnirelmann density]]
	*[[Shapley–Folkman lemma]]

	~~==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 \| 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.

	~~[[Category:Sumsets\|~~ ]]

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