“Kneser's theorem (combinatorics)”的意思、由来-开放百科全书

In mathematics, Kneser's theorem is an inequality among the sizes of certain sumsets in abelian groups. It belongs to the field of additive combinatorics, and is named after Martin Kneser, who published it in 1953.^[1] 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 is a prime number.^[2]

Statement

Let G be an abelian group and A, B finite non-empty subsets. If |A| + |B| ≤ |G| then there is a finite subgroup H of G such that^[3]

Notes

1. ^{{cite journal | first=Martin | last=Kneser | title=Abschätzungen der asymptotischen Dichte von Summenmengen | language=German | journal=Math. Z. | volume=58 | year=1953 | pages=459–484 | zbl=0051.28104 }}
2. ^¹{{harvtxt|Geroldinger|Ruzsa|2009|p=143}}
3. ^{{harvnb|Tao|Vu|2010|loc=pg. 200, Theorem 5.5}}

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|ref=harv }}

{{cite book | first=Melvyn B. | last=Nathanson | authorlink = Melvyn B. 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 }}

2 : Theorems in combinatorics|Sumsets