Topology of Cayley Graphs Applied to Inverse Additive Problems

Abstract

We present proofs of the basic isopermetric structure theory, obtaining some new simplified proofs. As an application, we obtain simple descriptions for subsets $S$ of an abelian group with $|kS|\le k|S|-k+1$ or $|kS-rS|-(k+r)|S|,$ where $1\le r\le k.$ These results may be applied to several questions in Combinatorics and Additive Combinatorics (Frobenius Problem, Waring's problem in finite fields and Cayley graphs with a big diameter, ....).

References in corpus (2)

• Hyper-atoms and the critical pair Theory
• A Structure Theory for Small Sum Subsets