On localization in Kronecker's diophantine theorem

Using a probabilistic approach, we extend for general $\mathbb Q$-linearly independent sequences a result of Turán concerning the sequence $( \log p_\ell)$, $p_\ell$ being the $\ell$-th prime. For instance let $ \lambda_1, \lambda_2, \ldots $ be linearly independent over $\mathbb Q$. We prove that there exists a constant $C_0$ such that for any positive integers $N $ and $\omega $, if $T > ({4\omega \over C_0} \sqrt{\log {N\omega\over C_0}})^{N} / \Xi $, where $$\Xi= \min_{{ u_k {\rm integers} \atop |u_k|\le 6 \omega \log {(N\omega/ C_0)}}\atop |u_1\lambda_1 + \ldots + u_N\lambda_N |\not = 0} |\sum_{1\le k \le N } \lambda_k u_k |$$ then to any reals $d, \beta_1,\ldots, \beta_N $, corresponds a real $t \in [d,d+T]$ such that $ \sup_{j=1}^N | t\l_j-\b_j | \le {1/ \omega}$.