On the Erd\H{o}s-Ulam problem in the plane : Erd\H{o}s-Ulam problem

In this paper we apply the method of compression to construct a dense set of points in the plane at rational distance from each other. We provide a positive solution to the Erd\H{o}s-Ulam problem本文应用压缩方法构造平面上的一个稠密点集，其中任意两点间的距离均为有理数。我们为厄尔多斯-乌拉姆问题（Erdős-Ulam problem）提供了一个肯定解。