Erdős-Ulam Problem

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

A set $A\subseteq{\mathbb{R}}^2$ is dense if every nonempty open Euclidean ball meets $A$. The distance requirement is $\|x-y\|_2\in{\mathbb{Q}}$ for every $x,y\in A$. Density is in the whole plane, not merely on a line or circle. No uniform lower bound on distances is imposed. [S1] [S2]

Statement recorded in the supplied source.

Is there a dense set of points in the plane with all pairwise distances rational? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Resolve the distribution of rational points on the general-type distance surfaces arising from four prescribed points, or find a fundamentally different construction of a plane-dense rational-distance set. [S1]

Short English statement

Can points be dense throughout the plane while every distance between two chosen points remains rational?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1279 (GEO-033), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 P. Corvaja, A. Turchet, and U. Zannier, Rational distances from given rational points in the plane, Geometriae Dedicata 219 (2025), article 59. https://doi.org/10.1007/s10711-025-01019-0. Bibliographic information from the supplied record.

• S3 J. Solymosi and F. de Zeeuw, On a question of Erdős and Ulam, Discrete & Computational Geometry 43 (2010), 393–401. https://arxiv.org/abs/0806.3095. Bibliographic information from the supplied record.

• S4 J. Shaffaf, A Solution of the Erdős–Ulam Problem on Rational Distance Sets Assuming the Bombieri–Lang Conjecture, Discrete & Computational Geometry 60 (2018), 87–95. https://arxiv.org/abs/1501.00159. Bibliographic information from the supplied record.

• S5 K. Ascher, L. Braune, and A. Turchet, The Erdős-Ulam problem, Lang's conjecture, and uniformity, Bulletin of the London Mathematical Society 52 (2020), 1053–1063. https://arxiv.org/abs/1901.02616. Bibliographic information from the supplied record.