On the distribution of (non)primitive lattice points

In 1959, Chalk and Erdös proved the following result on coprime inhomogeneous approximation to a real number: for any given $\alpha\in \mathbb{R}\setminus\mathbb{Q}$ and any real number $\eta$, there exists an absolute constant $ \lambda$ such that
$$ q\,|q\alpha - \eta - r| \leqslant \lambda\left(\frac{\log q}{\log\log q}\right)^2 $$
is satisfied by infinitely many pairs of coprime integers $q$, $r$ with $q\geqslant 1$.

In our lecture we discuss some recent related results and generalizations. In particular, we improve on a recent result by Svetlana Jitomirskaya and Wencai Liu.