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arXiv · 1901.02616

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

Kenneth Ascher·Lucas Braune·Amos Turchet
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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.

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Kenneth Ascher, Lucas Braune, Amos Turchet. 2019-01-09. The Erd\H{o}s-Ulam problem, Lang's conjecture, and uniformity. https://doi.org/10.1112/blms.12381

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