The Erdős–Ulam problem on rational distances

Is there a set of points that is dense in the plane and in which every pairwise distance is a rational number? Ulam asked this in 1946; Erdős believed the answer is no. A negative answer is known to follow from a deep conjecture in arithmetic geometry, but no unconditional proof is known.

Problem

A set $S\subseteq\mathbb{R}^2$ is a rational distance set if $\lvert x-y\rvert\in\mathbb{Q}$ for all $x,y\in S$. Is there a rational distance set that is dense in $\mathbb{R}^2$?

Background

Ulam posed the question in 1946, in response to the theorem of Anning and Erdős that an infinite set with all distances integral lies on a line. Rational distance sets that are dense in a line or in a circle are easy to build, so the question is about genuinely two-dimensional sets. Besicovitch conjectured that the limit points of a rational distance set cannot contain arbitrarily large convex sets. A 2014 blog post of Tao explains the connection with the Bombieri–Lang conjecture on rational points of varieties of general type.

What is known

• Conditional solution. Tao (2014, blog) and, independently, Shaffaf (2018) showed that, assuming the Bombieri–Lang conjecture, a rational distance set is contained in a finite union of real algebraic curves, and hence cannot be dense.
• Curves. Solymosi and de Zeeuw (2010) proved unconditionally that no irreducible algebraic curve other than a line or a circle contains an infinite rational distance set; as Tao notes, if a rational distance set is not Zariski dense, all but finitely many of its points lie on one line or one circle.
• General position. Ascher, Braune and Turchet (2020) showed, assuming Lang's conjecture, that rational distance sets in general position have uniformly bounded size.
• No unconditional result rules out a dense rational distance set.

What would count as progress

• An unconditional proof that no rational distance set is dense in the plane.
• Unconditional versions of the curve results for wider classes of sets.
• Large explicit rational distance sets in general position (the companion problem on integer distances asks for these); they can be verified by exact rational arithmetic.
• A formal proof against the Lean statement in formal-conjectures.

References

1. J. Solymosi and F. de Zeeuw, On a question of Erdős and Ulam, Discrete Comput. Geom. 43 (2010), 393–401. doi:10.1007/s00454-009-9179-x
2. T. Tao, The Erdős–Ulam problem, varieties of general type, and the Bombieri–Lang conjecture, blog post (2014). terrytao.wordpress.com
3. J. Shaffaf, A solution of the Erdős–Ulam problem on rational distance sets assuming the Bombieri–Lang conjecture, Discrete Comput. Geom. 60 (2018), 283–293. doi:10.1007/s00454-018-0003-3
4. K. Ascher, L. Braune and A. Turchet, The Erdős–Ulam problem, Lang's conjecture, and uniformity (2020). arXiv:1901.02616

Source

Imported from Erdős Problems 212, curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.