I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube.
Specifically: given a $d$-dimensional real vector, $\alpha = (\alpha_1,\ldots,\alpha_d) \in \mathbb{R}^d \cap [0,1]^d$, I would like to find a $d$-dimensional rational vector, $$\rho = \Bigg(\frac{p_1}{q},\ldots,\frac{p_d}{q} \Bigg) $$ where, $$ p_1,\ldots,p_d, q \in \mathbb{N}$$ $$ p_i \leq q ~~ \text{for} ~~ i = 1,\ldots,d$$ Note that the previous conditions simply ensure that $\rho \in\mathbb{Q}^d \cap [0,1]^d$
Given this setup, I am looking for an "error-bound" that is similar to Dirichlet's simultaneous approximation theorem. That is, something like:
Let $\alpha \in \mathbb{R}^d \cap [0,1]^d$. For every $\epsilon > 0$, we can choose an integer $q$ so that there exists integers $p_1,\ldots,p_d$ all less or equal to $q$ such that:
$$\Bigg|\alpha_i - \frac{p_i}{q} \Bigg| < \epsilon ~~~~ \text{for} ~~ i = 1,\ldots,d$$
Potentially helpful: from the answers to a previous question, I know that I can use Dirichlet's simultaneous approximation theorem to generate the following "there exists"-style error bound to my approximation problem:
Given any $\alpha \in \mathbb{R}^d \cap [0,1]^d$, there exists a natural number $N$ such that for all $q > N$, there exists integers $p_1,\ldots,p_d$ all less or equal to $q$ such that:
$$ \Bigg|\alpha_i - \frac{p_i}{q}\Bigg| < \frac{1}{q^{1+ 1/d}} \text{ for } i=1,\ldots,d $$
My issue with this bound is that it only guarantees that there exists some natural number $N$ for which my error bound holds... However it does not tell me how large $N$ should be. If someone can figure out how large $N$ should be then this would work as well.