3. Postdoctoral Program - MSRI

2 Main results
2.1 Line
Theorem 1. For any line L : ax + by + c = 0, for a, b, c ∈ Q and a �= 0 or b �= 0,
there exists a dense rational distance set with rational coordinates if and only if there
exists m ∈ Q such that a 2 + b 2 = m 2 .
Proof. Suppose L has a dense rational distance set with rational coordinates. Let
b �= 0, though we could equivalently let a �= 0, as the presiding condition is that a
and b are not simultaneously zero. Then we can express the line L as y = − a c x − b b .
Consider two points on L, (xi, yi) and (xj, yj). We can write the distance between
these two points as
using Lemma 1.
|xi − xj|
�
1 +
�
a
�2 b
The distance is rational only if 1 + a2
b 2 = n 2 for n ∈ Q. That is, if a 2 + b 2 = n 2 b 2 .
Hence, we get m = bn.
Suppose a 2 + b 2 = m 2 for m ∈ Q. Let b �= 0,then line L can be written as
y = − a c x − b b . The distance between two rational points on L, (xi, yi) and (xj, yj),
�
is |xi − xj| 1 + � �
a 2, 2 2 2 from Lemma 1. Note that we can write a + b = m as
b
1 + a2
b2 = m2
b2 �
. Hence we see 1 + a2
b2 = m,
which is rational.
b
3
(3)