Taking norms to be maximal norm, then the simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}} $,there are integers $${\displaystyle p_{1},\ldots ,p_{d},q\in \mathbb {Z} } $$ such that $$ {\displaystyle \left|\alpha _{i}-{\frac {p_{i}}{q}}\right|\leq {\frac {1}{q^{1+\frac{1}{d}}}}.}$$
Q1 : Can we expect to control the approximation to be near a given direction, i.e. given a point $P \in \mathbb{R}^d$, $\epsilon >0$ and a unit vector $ e \in S^{d-1}$, there is an integer $q$ and a point $Q \in \frac{1}{q}\mathbb{Z}^{d}$, such that$$ |P-Q| \leq \frac{C}{q^a}\ , \ \text{and}\quad \ |\frac{P-Q}{|P-Q|}-e|<\epsilon$$
