An inverse problem in number theory and geometric group theory

Abstract

This paper describes a new link between combinatorial number theory and geometry. The main result states that A is a finite set of relatively prime positive integers if and only if A = (K - K) ∩ N, where K is a compact set of real numbers such that for every x ε R there exists y ε K with x ≡ yx ≡ y (mod 1). In one direction, given a finite set A relatively prime positive integers, the proof constructs an appropriate compact set K such that A = (K - K) ∩ N. In the other direction, a strong form of a fundamental result in geometric group theory is applied to prove that (K - K) ∩ N is a finite set of relatively prime positive integers if K satisfies the appropriate geometrical conditions. Some related results and open problems are also discussed. © 2010 Springer Science+Business Media, LLC.