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 \(A = (K - K) \cap \mathbf{N}\), where K is a compact set of real numbers such that for every x ∈ R there exists y ∈ K with x ≡ yx ≡ ymod 1. In one direction, given a finite set A relatively prime positive integers, the proof constructs an appropriate compact set K such that \(A = (K - K) \cap \mathbf{N}\). 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.