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Source. N. H. Anning and P. Erdős, Integral distances, Bull. Amer. Math. Soc. 51 (1945), 598--600; the construction begins on p. 598 and continues on p. 599, and Ulam's question is on p. 599. The copy read is identified on the source card.

Read depth. Claims checked: the construction and the question were read clause by clause on the page images. The construction's two facts that the paper calls known (that its angle is an irrational multiple of $\pi$, and the resulting density) are cited there without proof and were not checked here. Nothing here is independently reviewed.

Statement

Construction (pp. 598--599, unnumbered). There is a set of points dense on the circle $x^2+y^2=1/4$ all of whose pairwise distances are rational. Before it (p. 598) the authors state that it is very likely that the points of their prime construction for the Theorem are dense on this circle, and that they cannot prove it.

Ulam's question (p. 599). Quoted, because it is a question as posed: "Several years ago Ulam asked whether it is possible to find a dense set in the plane such that all the distances are rational. We do not know the answer."

Construction

With $P_1=(-1/2,0)$ and $P_2=(1/2,0)$, take the point $X_1$ of the circle at distance $3/5$ from $P_1$, so at distance $4/5$ from $P_2$, and let $\alpha$ be the angle $P_2P_1X_1$, so that $\sin\alpha$ and $\cos\alpha$ are rational. The paper states that $\alpha$ is known to be an irrational multiple of $\pi$. The points $X_i$ of the circle with angle $P_1P_2X_i$ equal to $i\alpha$ are then dense on the circle, and their mutual distances are rational because $\sin i\alpha$ and $\cos i\alpha$ are rational.

Two misprints on p. 599 leave the construction unchanged: the point named as at distance $4/5$ from $X_1$ is printed as $(0,1/2)$, where the construction uses $P_2=(1/2,0)$, and the circle on which the $X_i$ are dense is printed as $X^2+y^2=1/2$, where the construction's circle is $x^2+y^2=1/4$.

Dependencies

Outside the paper: that $\alpha$ is an irrational multiple of $\pi$, which the paper calls known, and the density of the multiples of an irrational rotation on the circle.

Bears on

• Problem 212: the problem's statement is the question the paper records from Ulam (p. 599), and the paper states that the authors do not know the answer. The construction gives a rational-distance set dense on a circle, not dense in the plane, and answers nothing about the problem.