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Zbl 1143.11039
Balandraud, Éric
A variant of the isoperimetric method of Hamidoune, applied to Kneser's theorem. (Une variante de la méthode isopérimétrique de Hamidoune, appliquée au théorème de Kneser.) (French)
[J] Ann. Inst. Fourier 58, No. 3, 915-943 (2008). ISSN 0373-0956; ISSN 1777-5310/e

In additive number theory, a central result is Kneser's theorem: Let $G$ be an abelian group, let $A, B \subset G$ be two finite subsets and suppose that $|A + B| < |A|+|B|-1$. Then there exists a subgroup $H \subset G$ such that $A + B + H = A + B$, and if we choose a maximal such $H$, then $|A + B| = |A+H|+|B+H|-|H|$. More recently, Hamidoune developed a different approach to additive problems, the ``isoperimetric'' method; using this he was able to generalize several classical results. However, no proof of Kneser's theorem using the isoperimetric method was known. The present article describes a variant of the isoperimetric method which does yield a proof of Kneser's theorem. The main ideas are the following. Fix once and for all a set $B\subset G$, and for simplicity let us suppose $0 \in B$. We want to understand the map which sends a set $X \subset G$ to $X + B$ and in particular how much the set grows: $\Phi_B(X) := |(X + B) \smallsetminus X|$. A key definition is the notion of ``cells'': sets $C$ such that for any $X \supsetneq C$ we have $X + B \supsetneq C + B$. The author develops a whole theory around them, partly even for non-abelian groups $G$. Among others, there is a duality between cells for $B$ and cells for $-B$, given by $C \mapsto G \smallsetminus (C + B)$. The main structural result is the following. Fix $\lambda < |B| - 1$ such that cells $C$ with $\Phi_B(C) = \lambda$ do exist. Then there exists a subgroup $N \subset G$ such that for any cell $C$ with $\Phi_B(C) = \lambda$ we have $C + N = C$, and moreover $N$ itself is a cell with $\Phi_B(N) = \lambda$. (The name ``isoperimetric'' comes from the fact that $\Phi_B(C)$ is the perimeter of $C$ in the Cayley graph of $(G, B)$.)
[Immanuel Halupczok (Paris)]

MSC 2000:: *11P70 Inverse problems of additive number theory
11B75 Combinatorial number theory
20D60 Arithmetic and combinatorial problems on finite groups

Keywords: additive problems; sumsets; isomerimetric method; additive number theory; Kneser's theorem

Cited in: Zbl 1231.11124 Zbl 1197.11016

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