# Rational points on the unit circle

Is anything known about any of the following questions about rational points on the unit circle? By “double point” I mean an element of $2C$, where $C$ is the group of rational points on the unit circle (i.e. Gaussian rationals with unit modulus, under multiplication).

1. Are there two rational points $(x_1, y_1)$ and $(x_2, y_2)$ on the unit circle such that the numbers $y_1 + y_2$ and $y_1 - y_2$ are also the y-coordinates of rational points on the unit circle, except where $y_1$ or $y_2$ is zero?

2. Are there are three distinct double points on the unit circle whose y-coordinates are in arithmetic progression, other than the trivial case where the middle one is on the x-axis?

3. Are there two double points $(x_1, y_1)$ and $(x_2, y_2)$ on the unit circle such that $y_1 + y_2$ is also the y-coordinate of a double point on the unit circle, excluding the trivial case where one of $y_1$, $y_2$, $y_1+y_2$ is zero?

4. Are there three distinct rational points on the unit circle whose y-coordinates are in arithmetic progression, where the common difference of this arithmetic progression is also the y-coordinate of a rational point on the unit circle, other than the trivial case where the middle point is on the x-axis?