The Erdős-Ulam problem, Lang's conjecture, and uniformity

Abstract

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.

arXiv comment9 pages. Improved exposition throughout. Version to appear in the Bulletin of the London Math Society

References in corpus (3)

• On a question of Erdos and Ulam
• A solution of the Erdos-Ulam problem on rational distance sets assuming the Bombieri-Lang conjecture
• Hyperbolicity and uniformity of varieties of log general type

Cited by in corpus (1)

• Rational distances from given rational points in the plane