Is there a dense of points in the real plane so that every two points are at a rational distance

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.

For $x=(x_1,x_2)$ and $y=(y_1,y_2)$ in $\mathbb R^2$, use Euclidean distance \[|x-y|=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}.\] A subset $S\subset\mathbb R^2$ is dense if every nonempty Euclidean open disc meets $S$. It is a rational-distance set if $|x-y|\in\mathbb Q$ for every $x,y\in S$.

Statement recorded in the supplied source.

Does there exist a dense rational-distance set $S\subset\mathbb R^2$? [S2]

Short English statement

Can a set be dense throughout the real plane while every pair of its points is a rational distance apart?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100066 (AMR-030-0066). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jozsef Solymosi and Frank de Zeeuw, On a question of Erdős and Ulam (2009 revision), Abstract and Section1, definition of rational set and Ulam’s everywhere-dense-plane question, page1. https://arxiv.org/abs/0806.3095.