The Erd\H{o}s-Ulam problem, Lang's conjecture, and uniformity
Number Theory
2020-08-19 v2 Algebraic Geometry
Combinatorics
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.
Keywords
Cite
@article{arxiv.1901.02616,
title = {The Erd\H{o}s-Ulam problem, Lang's conjecture, and uniformity},
author = {Kenneth Ascher and Lucas Braune and Amos Turchet},
journal= {arXiv preprint arXiv:1901.02616},
year = {2020}
}
Comments
9 pages. Improved exposition throughout. Version to appear in the Bulletin of the London Math Society