Content deleted Content added
m clarifed which Kneser |
added some in-line citations, reference, expanded statement of theorem |
||
Line 1:
In mathematics, in the field of [[additive combinatorics]], '''Kneser's theorem''', named after [[Martin Kneser]], is a statement about [[Sumset|set addition]] in [[finite group]]s.<ref>M. Kneser, Abschätzungen der asymptotischen Dichte von Summenmengen, ''Math. Z.'', '''58''' (1953), 459-484.</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'' of ''A''+''B''
:<math> H = \lbrace g \in G : g + (A+B) = (A+B) \rbrace . </math>
==Notes==
{{reflist}}
==References==
* {{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|first2=Van H.|last2=Vu|title=Additive Combinatorics|year=2010|publisher=Cambridge University Press|place=Cambridge|isbn=978-0-521-13656-3}}
[[Category:Theorems in combinatorics]]
| |||