### 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.

Original language | English (US) |
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Journal | Bulletin of the London Mathematical Society |

DOIs | |

State | Accepted/In press - 2020 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- 14G05
- 14J17
- 14J29 (secondary)
- 52C10 (primary)

## Fingerprint Dive into the research topics of 'The Erdős–Ulam problem, Lang's conjecture and uniformity'. Together they form a unique fingerprint.

## Cite this

Ascher, K., Braune, L., & Turchet, A. (Accepted/In press). The Erdős–Ulam problem, Lang's conjecture and uniformity.

*Bulletin of the London Mathematical Society*. https://doi.org/10.1112/blms.12381