Dense rational distance sets in the plane

Importance: High ✭✭✭
Author(s): Ulam, Stanislaw M.
Subject: Geometry
Recomm. for undergrads: no
Posted by: mdevos
on: July 4th, 2008

\begin{problem} Does there exist a dense set $S \subseteq {\mathbb R}^2$ so that all pairwise distances between points in $S$ are rational? \end{problem}

This famous problem was asked by Ulam, who guessed the answer would be negative.

A cute theorem of Erdos shows that if $S \subseteq {\mathbb R}^2$ is non-collinear and all pairwise distances between points in $S$ are integral, then $S$ is finite. For the proof, first note that if $x,y \in {\mathbb R}^2$ have distance $k \in {\mathbb Z}$, then every point which has integer distance to both $x$ and $y$ must lie on one of the $k+1$ hyperbolas consisting of those $z \in {\mathbb R}^2$ with $|{\mathit dist}(x,z) - {\mathit dist}(y,z)| = j$ for some $0 \le j \le k$. So, if all pairwise distances between points in $S$ are integral, and $x,y,z \in S$ are non-collinear, then every other point in $S$ must lie on an intersection between one of finitely many hyperbola with foci $x,y$ and one of finitely many with foci $x,z$. This set is necessarily finite, thus completing the proof.

Of course, the above argument gives no upper bound on the size of a non-collinear set of points in ${\mathbb R}^2$ with pairwise integral distances. Indeed, if Ulam's conjecture is true, then there exist such sets of arbitrary size. Surprisingly, it is very difficult to construct such sets $S$ of even rather small size. Recently Kreisel and Kurz [KK] found such a set of size 7, but it is unknown if there exists one of size 8.

It is trivial to find infinitely many points on a line with all pairwise distances rational. Less trivially, there exist infinite subsets of a circle with all pairwise distances rational. Very recently, Solymosi and De Zeeuw [SZ] proved that these are the only two irreducible algebraic curves with this property. This suggests that, if the answer to Ulam's problem is affirmative, such a set $S$ must be extremely special.

Bibliography

[KK] T. Kreisel and S. Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete & Computational Geometry, Online first: DOI 10.1007/s00454-007-9038-6

[SZ] J. Solymosi and F. de Zeeuw, \href[On a question of Erdos and Ulam]{http://arxiv.org/PS_cache/arxiv/pdf/0806/0806.3095v1.pdf}.


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