Rational distance sets and Lang’s Conjecture

A rational distance set is a subset of the real plane in which every two points have rational distance. A famous question of Ulam, based on work of Anning and Erdös, asks whether any such set can be dense for the Euclidean topology. Tao and Shaffaf showed that Lang's Conjecture implies that a rational distance set can never be dense. We show, following works of Solymosi, de Zeeuw, Makhul and Shaffaf that one can in fact derive from Lang’s Conjecture bounds for the size of rational distance sets in general position. This is based on joint work with K. Ascher and L. Braune.