Geodesic
Asymptotic number of solutions of some systems of diophantine inequalities
Matematičeskie zametki, Tome 11 (1972) no. 6, pp. 619-623 
V. I. Bernik. Asymptotic number of solutions of some systems of diophantine inequalities. Matematičeskie zametki, Tome 11 (1972) no. 6, pp. 619-623. http://geodesic.mathdoc.fr/item/MZM_1972_11_6_a1/
@article{MZM_1972_11_6_a1,
     author = {V. I. Bernik},
     title = {Asymptotic number of solutions of some systems of diophantine inequalities},
     journal = {Matemati\v{c}eskie zametki},
     pages = {619--623},
     year = {1972},
     volume = {11},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_11_6_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\lambda\geqslant0$.