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) \cap N, where K is a compact set of real numbers such that for every real number x there exists y in K with x \equiv y mod 1. In one direction, given a finite set A of relatively prime positive integers, the proof constructs an appropriate compact set K such that A = (K-K) \cap N. In the other direction, a strong form of a fundamental theorem in geometric group theory is applied to prove that (K-K)\cap 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.