An inverse problem in number theory and geometric group theory

Details An inverse problem in number theory and geometric group theory An inverse problem in number theory and geometric group theory

2009-01-11

arxiv.org/abs/0901.1458v2

Mathematics

Number Theory

8 pages, minor revisions and corrections

Scientific paper

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.

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