An inverse problem in number theory and geometric group theory

TL;DR

This paper establishes a novel connection between combinatorial number theory and geometry by characterizing finite sets of relatively prime positive integers through properties of certain compact sets and geometric conditions.

Contribution

It introduces a new characterization of finite relatively prime sets using geometric and topological methods, linking number theory with geometric group theory.

Findings

Finite sets of relatively prime positive integers correspond to differences of specific compact sets.

A geometric group theory theorem is used to prove the characterization.

The paper discusses related results and open problems in the field.

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.