# An inverse problem in number theory and geometric group theory

- Authors
- Type
- Published Article
- Publication Date
- Jun 01, 2009
- Submission Date
- Jan 11, 2009
- Source
- arXiv
- License
- Yellow
- External links

## 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.