Erdős–Ulam dense rational distance set problem

Determine whether the Euclidean plane contains a dense rational distance set.

Background

A rational distance set is a subset of Euclidean space in which every distance between two points is rational. The paper distinguishes the existence of infinite rational distance sets from the stronger question of metric density. It notes that the classical Erdős–Ulam problem asks whether the plane contains a dense rational distance set and remains unresolved.

The paper’s constructions produce countably infinite rational distance sets with strong affine or spherical nondegeneracy properties in every dimension, but they do not address density in the plane. Thus the Erdős–Ulam problem is an explicitly stated unresolved question outside the scope of the paper’s main construction results.