Dense rational distance sets in the plane

 Importance: High ✭✭✭
 Author(s): Ulam, Stanislaw M.
 Subject: Geometry
 Keywords: integral distance rational distance
 Posted by: mdevos on: July 4th, 2008
Problem   Does there exist a dense set so that all pairwise distances between points in are rational?

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

A cute theorem of Erdos shows that if is non-collinear and all pairwise distances between points in are integral, then is finite. For the proof, first note that if have distance , then every point which has integer distance to both and must lie on one of the hyperbolas consisting of those with for some . So, if all pairwise distances between points in are integral, and are non-collinear, then every other point in must lie on an intersection between one of finitely many hyperbola with foci and one of finitely many with foci . 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 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 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 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, On a question of Erdos and Ulam.

* indicates original appearance(s) of problem.