  A rational distance set is a subset of the plane such that the distance
between any two points is a rational number. We show, assuming Lang's
Conjecture, that the cardinalities of rational distance sets in general
position are uniformly bounded, generalizing results of Solymosi-de Zeeuw,
Makhul-Shaffaf, Shaffaf, and Tao. In the process, we give a criterion for
certain varieties with non-canonical singularities to be of general type.
