I am work with something on diophantine approximation, and I fine a high dimensional generation of Dirichlet approximation theorem seems to be true, I will be very happy if this is true. the statement is following:
>**Dirichlet approximation theorem, high dimensional generation**
Let $\alpha_1,..,\alpha_n\in \mathbb R$ be n linear independent numbers on $\mathbb Q$. Then $\forall x\notin \mathbb Q(\alpha_1,...,\alpha_n)$, there is infinite many tuples $c-(c_1,...,c_n)\in \mathbb Q^n$ such that,
$$|x-\sum_{i=1}^n c_i\alpha_i|<\frac{1}{\|c\|^{n+1}_{hight}}$$
Where $\|c\|_{hight}:=\max_i \{\|c_i\|_{hight}\}$, $\|c_i\|_{hight}=\max {\{|p|,|q|, c_i=\frac{q}{p}, (p,q)=1\}}$.

It is not difficult to see this conjecture coincide with the classical dirichlet problem when $n=1$.

Have this problem been studied? Do to the problem is so natural I think it should be a classical result but I can not find it in classical books in diophantine approximation.

I will appreciate for point out a reference or some advice, thanks!