The Erdős–Ulam problem, Lang's conjecture and uniformity

Kenneth Ascher, Lucas Braune, Amos Turchet

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 languageEnglish (US)
Pages (from-to)1053-1063
Number of pages11
JournalBulletin of the London Mathematical Society
Issue number6
StatePublished - Dec 2020

All Science Journal Classification (ASJC) codes

  • General Mathematics


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


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