3. Postdoctoral Program - MSRI

In 1945 Stanislaw Ulam posed a question about a rational distance set: is there
an everywhere dense rational distance set in the plane[3]? Paul Erdos also considered
the problem and he conjectured that the only irreducible algebraic curves which have
an infinite rational distance set are the line and the circle[2]. This was later proven
to be true by Jozsef Solymosi and Frank de Zeeuw[2].
First, we provide the formal definition of a rational distance set:
Definition 1. A rational distance set S is a set of elements Pi = (xi, yi) ∈ R 2 ,
1 ≤ i ≤ n, n ∈ Z, such that for all Pi, Pj ∈ S, i �= j, ||Pi − Pj|| is a rational number.
We are primarily concerned with finding rational distance sets of rational points.
The distance between two points on a graph is � (xi − xj) 2 + (yi − yj) 2 . It will prove
useful to rewrite this formula as
|xi − xj|
Now, consider the following lemma:
Lemma 1. The rational solutions to the equation
can be parametrized by α = m2 +1
�
1 + ( yi − yj
)
xi − xj
2 (1)
α 2 = 1 + β 2
2m , β = m2−1 2m
For the proof, refer to Campbell’s paper [1].
2
, for m ∈ Q, m �= 0.
(2)