Rational translates of the unit circle cover the plane Is it true that the translations of the unit circle by vectors
with both coordinates rational cover the plane?
This comes to solving
$$ x=a+\cos \theta, \ y=b+\sin \theta$$
with unknowns $a,b$ rational and $\theta$
between $0,2\pi$.
I couldn't find a positive answer or an immediate contradiction.
(I typed the question from a phone
so if I made new tags by error, please correct them)

No, the above statement is false.This is because it would imply that every real number is algebraic over $\mathbb{Q}$.
