Erdős–Ulam problem

In mathematics, the Erdős–Ulam problem asks whether the plane contains a dense set of points whose Euclidean distances are all rational numbers. It is named after Paul Erdős and Stanislaw Ulam.

Unsolved problem in mathematics: Is there a dense set of points in the plane at rational distances from each other? (more unsolved problems in mathematics)

Large point sets with rational distances

The Erdős–Anning theorem states that a set of points with integer distances must either be finite or lie on a single line.[1] However, there are other infinite sets of points with rational distances. For instance, on the unit circle, let S be the set of points

${\displaystyle (\cos \theta ,\sin \theta )}$

where ${\displaystyle \theta }$ is restricted to values that cause ${\displaystyle \tan {\tfrac {\theta }{4}}}$ to be a rational number. For each such point, both ${\displaystyle \sin {\tfrac {\theta }{2}}}$ and ${\displaystyle \cos {\tfrac {\theta }{2}}}$ are themselves both rational, and if ${\displaystyle \theta }$ and ${\displaystyle \varphi }$ define two points in S, then their distance is the rational number

${\displaystyle \left|2\sin {\frac {\theta }{2}}\cos {\frac {\varphi }{2}}-2\sin {\frac {\varphi }{2}}\cos {\frac {\theta }{2}}\right|.}$

More generally, a circle with radius ${\displaystyle \rho }$ contains a dense set of points at rational distances to each other if and only if ${\displaystyle \rho ^{2}}$ is rational.[2] However, these sets are only dense on their circle, not dense on the whole plane.

History and partial results

In 1946, Stanislaw Ulam asked whether there exists a set of points at rational distances from each other that forms a dense subset of the Euclidean plane.[2] While the answer to this question is still open, József Solymosi and Frank de Zeeuw showed that the only irreducible algebraic curves that contain infinitely many points at rational distances are lines and circles.[3] Terence Tao and Jafar Shaffaf independently observed that, if the Bombieri–Lang conjecture is true, the same methods would show that there is no infinite dense set of points at rational distances in the plane.[4][5] Using different methods, Hector Pasten proved that the abc conjecture also implies a negative solution to the Erdős–Ulam problem.[6]

Consequences

If the Erdős–Ulam problem has a positive solution, it would provide a counterexample to the Bombieri–Lang conjecture and to the abc conjecture. It would also solve Harborth's conjecture, on the existence of drawings of planar graphs in which all distances are integers. If a dense rational-distance set exists, any straight-line drawing of a planar graph could be perturbed by a small amount (without introducing crossings) to use points from this set as its vertices, and then scaled to make the distances integers. However, like the Erdős–Ulam problem, Harborth's conjecture remains unproven.

References

1. Anning, Norman H.; Erdős, Paul (1945), "Integral distances", Bulletin of the American Mathematical Society, 51 (8): 598–600, doi:10.1090/S0002-9904-1945-08407-9.
2. Klee, Victor; Wagon, Stan (1991), "Problem 10 Does the plane contain a dense rational set?", Old and New Unsolved Problems in Plane Geometry and Number Theory, Dolciani mathematical expositions, 11, Cambridge University Press, pp. 132–135, ISBN 978-0-88385-315-3.
3. Solymosi, József; de Zeeuw, Frank (2010), "On a question of Erdős and Ulam", Discrete and Computational Geometry, 43 (2): 393–401, arXiv:0806.3095, doi:10.1007/s00454-009-9179-x, MR 2579704
4. Tao, Terence (2014-12-20), "The Erdos-Ulam problem, varieties of general type, and the Bombieri-Lang conjecture", What's new, retrieved 2016-12-05
5. Shaffaf, Jafar (May 2018). "A Solution of the Erdős–Ulam Problem on Rational Distance Sets Assuming the Bombieri–Lang Conjecture". Discrete and Computational Geometry. 60 (8). arXiv:1501.00159. doi:10.1007/s00454-018-0003-3.
6. Pasten, Hector (2017), "Definability of Frobenius orbits and a result on rational distance sets", Monatshefte für Mathematik, 182 (1): 99–126, doi:10.1007/s00605-016-0973-2, MR 3592123