A variant of the isoperimetric method of Hamidoune, applied to Kneser's theorem
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 \((G, B)\).)
scientific article; zbMATH DE number 5298326
scientific article