A variant of the isoperimetric method of Hamidoune, applied to Kneser's theorem
type: Item
instance of: scholarly article
title: A variant of the isoperimetric method of Hamidoune, applied to Kneser's theorem
zbMATH Open document ID: 1143.11039
author: Éric Balandraud
publication date: 2008-07-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10339
review text: 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 \((G, B)\).)
reviewed by: Immanuel Halupczok
Mathematics Subject Classification ID: 11P70
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 5298326
zbMATH Keywords: additive problems
zbMATH Keywords: sumsets
zbMATH Keywords: isomerimetric method
zbMATH Keywords: additive number theory
zbMATH Keywords: Kneser's theorem
MaRDI profile type: Publication
OpenAlex ID: W2314825613
cites work: On a product of finite subsets in a torsion-free group
cites work: Q3845442
cites work: On the Addition of Residue Classes
cites work: A Historical Note
cites work: On Kneser's Addition Theorem in Groups
cites work: Q3275020
cites work: Handbook of Graph Theory
cites work: Q4121925
cites work: On the connectivity of Cayley digraphs
cites work: An isoperimetric method in additive theory
cites work: Subsets with small sums in abelian groups. I: The Vosper property
cites work: Q4248195
cites work: Some results in additive number theory I: The critical pair theory
cites work: A generalization of Freiman's 3k-3 Theorem
cites work: A multiple set version of the \(3k-3\) theorem
cites work: On small sumsets in an abelian group
cites work: Abschätzung der asymptotischen Dichte von Summenmengen
cites work: Ein Satz über abelsche Gruppen mit Anwendungen auf die Geometrie der Zahlen
cites work: An addition theorem of Abelian groups for sets of elements
cites work: Q4889853
cites work: On Erdős and Graham's \(X\) function.
cites work: The Critical Pairs of Subsets of a Group of Prime Order
cites work: A generalisation to noncommutative groups of a theorem of Mann
DOI: 10.5802/AIF.2374
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5298326