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This paper describes a new link between combinatorial number theory andgeometry. The main result states that A is a finite set of relatively primepositive integers if and only if A = (K-K) \cap N, where K is a compact set ofreal 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 primepositive integers, the proof constructs an appropriate compact set K such thatA = (K-K) \cap N. In the other direction, a strong form of a fundamentaltheorem in geometric group theory is applied to prove that (K-K)\cap N is afinite set of relatively prime positive integers if K satisfies the appropriategeometrical conditions. Some related results and open problems are alsodiscussed.

An inverse problem in number theory and geometric group theory