数论和几何群论中的一个反问题
An inverse problem in number theory and geometric group theory
摘要：
本文描述了组合数论与几何之间的一种新的联系。主要结果是：A是互素正整数的有限集当且仅当A =（K-K）\cap N，其中K是紧实数集，使得对每个实数x，在K中存在y，x \equiv y mod 1.在一个方向上，给定一个互素正整数的有限集A，证明构造一个适当的紧集合K，使得A =（K-K）\cap N。另一方面，应用几何群论中一个基本定理的强形式，证明了当K满足适当的几何条件时，（K-K）\cap N是有限的互素正整数集.并讨论了一些相关结果和尚待解决的问题。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
