Consider an $\alpha \in \mathbb{R} \setminus \mathbb{Q}$.
When doing diophantine approximation in one dimension, a result of Lagrange gives the structure of the $p, q \in \mathbb{Z}$ that verify $$ | q \alpha - p | < \frac{1}{2q}. $$ In fact, they are exactly the convergents (truncated continued fractions) of $\alpha$.
My question is about simultaneous approximation.
Considering an $\alpha \in \mathbb{R}^n$ such that $1, \alpha_1, \dots, \alpha_n$ are linearly independent over the rationals, can we say something on the structure of the solutions $q \in \mathbb{N}, p \in \mathbb{Z}^n$ such that $$ \| q \alpha - p \|_\infty < \frac{c}{q^{1/n}} ?$$
My second question is about Schmidt's theorem. It implies that, for any $\epsilon > 0$, there exist only finitely many integer solutions of the system $$ \| q \alpha - p \|_\infty < \frac{1}{q^{1/n + \epsilon}}.$$ This means that there exists a constant $c$ such that $$ \| q \alpha - p \|_\infty < \frac{c}{q^{1/n + \epsilon}}$$ admits no integer solution. In the case that $\mathbb{Q}(\alpha_1, \dots, \alpha_n)$ has degree $n+1$ over $\mathbb{Q}$ and $\alpha_1$ is an algebraic integer, we can show that $\alpha$ is badly approximable and give a lower-bound on $c$.
Are there any similar results, bounding the constant $c = c(\alpha, \epsilon)$, if we drop the condition that the $\alpha_i$ are conjugate elements? For example, is the vector $\alpha := (\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7})$ badly approximable? Is the constant $c(\alpha, 1/n)$ bounded away from zero?
Thanks!