Kneser's theorem (combinatorics)

In mathematics, in the field of additive combinatorics, Kneser's theorem, named after Martin Kneser, is a statement about set addition in finite groups.

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

${\begin{aligned}|A+B|&\geq |A+H|+|B+H|-|H|\\&\geq |A|+|B|-|H|.\end{aligned}}$

The subgroup H can be taken to be the stabiliser of A+B

$H=\lbrace g\in G:g+(A+B)=(A+B)\rbrace .$

References

• Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer-Verlag. pp. 109–132. ISBN 0-387-94655-1. Zbl 0859.11003.
• Tao, Terence; Vu, Van H. (2010), Additive Combinatorics, Cambridge: Cambridge University Press, ISBN 978-0-521-13656-3