Asymptotic number of solutions of some systems of diophantine inequalities

The problem of finding the asymptotic number of solutions of the system of inequalities
\begin{gather*} ||\alpha_iq||<q^{-\sigma_i}\qquad(i=1,\dots,n),\quad\sigma_i>0,\\ \sigma=\sum_{i=1}^n\sigma_i<c(\alpha_1,\dots,\alpha_n),\qquad q=1,\dots,N,\\ \end{gather*}
is solved under the assumption that for real numbers $\alpha_1,\dots,\alpha_n$, starting from some $Q=\max(q_1,\dots,q_n)$ the inequality
$$ ||\alpha_1q_1+\dots+\alpha_nq_n||\geqslant\frac1{Q^{n+\lambda}} $$
holds for any real $\lambda\geqslant0$.