Some additive applications of the isoperimetric approach

Hamidoune, Yahya O.

Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2007-2036.

Soient $G$ un groupe et $X$ un sous-ensemble fini de $G$. La méthode isopérimétrique étudie la fonction objective $\left|\right(XB)\setminus X|$, définie sur les parties $X$ telles que $\left|X\right|\ge k$ et $|G\setminus (XB\left)\right|\ge k$, où $XB$ est le produit de $X$ par $B$. Les inégalités additives découlent de la structure des ensembles où cette fonction atteint sa valeur minimale.

Nous présentons dans ce mémoire les bases de cette méthode et certaines de ses applications. Nous obtenons quelques nouveaux résultats et des courtes preuves de résultats connus.

Certains des résultats obtenus dans ce travail seront appliqués dans un futur mémoire afin d’améliorer les théorèmes de structure de Kempermann.

Let $G$ be a group and let $X$ be a finite subset. The isoperimetric method investigates the objective function $\left|\right(XB)\setminus X|$, defined on the subsets $X$ with $\left|X\right|\ge k$ and $|G\setminus (XB\left)\right|\ge k$, where $XB$ is the product of $X$ by $B$.

In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and obtain generalizations and short proofs for several known results. We also give some new applications.

Some of the results obtained here will be used in coming papers to improve Kempermann structure Theory.

Classification : 05C25, 20D60, 11B75, 05C40
Keywords: Addition theorem, Cayley graph, inverse additive theory
Mot clés : somme de Minkowski, graphe de Cayley, problème inverse

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