On the generalized Roth–Schmidt theorem

It is proved that the inequality
$$ \prod_{i=1}^{n-1}\|q\theta_i\|<c(qf(q))^{-1}, $$
where $c$ is a fixed constant, $f(q)>\log q$ and $\theta_1,\dots,\theta_{n-1}$ belong to a totally real algebraic number field of degree $n$ can be solved for arbitrary large $q$. For $n=3$ necessary and sufficient conditions are given in order that $f(q)=O(\log q)$.