# The Erd\H{o}s-Ulam problem, Lang's conjecture, and uniformity

## TL;DR

This paper demonstrates, assuming Lang's Conjecture, that rational distance sets in the plane in general position have a uniform upper bound on their size, extending previous results and providing a criterion for certain algebraic varieties to be of general type.

## Contribution

It establishes a conditional uniform bound on rational distance sets assuming Lang's Conjecture and introduces a criterion for varieties with non-canonical singularities to be of general type.

## Key findings

- Under Lang's Conjecture, rational distance sets are uniformly bounded in size.
- Provides a new criterion for varieties with non-canonical singularities to be of general type.
- Generalizes previous results on rational distance sets in the plane.

## 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.