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)

There are only countably many rational translates of the unit circle $S^1\subset{\mathbb R}^2$. Altogether they make up a set of measure zero.
