Dense rational distance sets in the plane

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Distances are Euclidean: $\|x-y\|_2=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$ for $x,y\in\mathbb R^2$. A set $S$ is dense in the plane if every open disk of positive radius meets $S$, equivalently $\overline S=\mathbb R^2$.

Statement recorded in the supplied source.

Is there $S\subseteq\mathbb R^2$ such that \[\overline S=\mathbb R^2\quad\text{and}\quad\|x-y\|_2\in\mathbb Q\quad\text{for every }x,y\in S?\] [S2]

Short English statement

Can a set dense throughout the plane have rational Euclidean distances between every pair of its points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3077 (OPG-1761). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ulam, Stanislaw M., Dense rational distance sets in the plane, Open Problem Garden, node 1761. Complete problem, discussion and bibliography (including mathematical image alt text). https://www.openproblemgarden.org/op/dense_rational_distance_sets_in_the_plane.