Topology of Cayley Graphs Applied to Inverse Additive Problems

TL;DR

This paper develops new simplified proofs for isoperimetric structures in Cayley graphs and applies these results to characterize subsets in abelian groups with specific sumset properties, impacting combinatorics and additive number theory.

Contribution

It provides new simplified proofs of isoperimetric structure theory and applies these to describe subsets with particular sumset bounds in abelian groups.

Findings

Simplified proofs of isoperimetric structure theory.

Descriptions of subsets with small sumsets in abelian groups.

Applications to combinatorics and additive number theory 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, ....).