Skew and direct sums of permutations: Difference between revisions

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In mathematics, '''Kneser's theorem''' is an [[Inequality (mathematics)|inequality]] among the sizes of certain [[sumset]]s in [[finite abelian group]]s. It belongs to the field of [[additive combinatorics]], and is named after [[Martin Kneser]], who published it in 1953.<ref>{{cite journal | first=Martin | last=Kneser | title=Abschätzungen der asymptotischen Dichte von Summenmengen | language=German | journal=[[Math. Zeitschr.]] | volume=58 | year=1953 | pages=459–484 | zbl=0051.28104 }}
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</ref>  It may be regarded as an extension of the [[Cauchy–Davenport theorem]], which also concerns sumsets in groups but is restricted to groups whose [[Order (group theory)|order]] is a [[prime number]].<ref name=GR143>Geroldinger & Rusza (2009) p.143</ref>
 
==Statement==
Let ''G'' be a non-trivial [[abelian group]] and ''A'', ''B'' finite non-empty subsets. If |''A''| + |''B''| ≤ |''G''| then there is a finite subgroup ''H'' of ''G'' such that<ref>{{harvnb|Tao|Vu|2010|loc=pg. 200, Theorem 5.5}}</ref>
 
:<math>\begin{align} |A+B| &\ge |A+H| + |B+H| - |H| \\ &\ge |A| + |B| - |H|. \end{align} </math>
 
The subgroup ''H'' can be taken to be the ''stabiliser''<ref name=GR143/> of ''A''+''B''
 
:<math> H = \lbrace  g \in G : g + (A+B) = (A+B) \rbrace. </math>
 
==Notes==
{{reflist}}
 
==References==
* {{cite book | editor1-last=Geroldinger | editor1-first=Alfred | editor2-last=Ruzsa | editor2-first=Imre Z. | editor2-link = Imre Z. Ruzsa | others=Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse) | title=Combinatorial number theory and additive group theory | series=Advanced Courses in Mathematics CRM Barcelona | location=Basel | publisher=Birkhäuser | year=2009 | isbn=978-3-7643-8961-1 | zbl=1177.11005 }}
* {{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 | pages=109–132 }}
* {{citation | first1=Terence | last1=Tao | author1-link=Terence Tao | first2=Van H. | last2=Vu | title=Additive Combinatorics | year=2010 | publisher=[[Cambridge University Press]] | place=[[Cambridge]] | isbn=978-0-521-13656-3 | zbl=1179.11002 }}
 
[[Category:Theorems in combinatorics]]
[[Category:Sumsets]]
 
{{combin-stub}}

Latest revision as of 21:48, 5 June 2014

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