Let $\mathbf{x} \in \mathbb{R}^K$ be an irrational vector. Assume that $\|\mathbf{x}\|^2 \leq 1$. Is is known that for all $N > 1$, there exists an $p_1 \in \mathbb{N}, \mathbf{q}_1 \in \mathbb{N}^{K}$ such that $|p_1| < N$ and

\begin{equation} \|p_1\mathbf{x} - \mathbf{q}_1\| \leq \frac{C}{N^{\frac{1}{K}}} \end{equation}

where $C$ is some constant.

I want to find another K-1 pairs $(p_2, \mathbf{q}_2)$, .., $(p_K, \mathbf{q}_K)$ where $\mathbf{p}_1,..,\mathbf{p}_K \in \mathbf{N}^K$ are linearly independent and such that

\begin{equation} \max_i \|p_i\mathbf{x} - \mathbf{q}_i\| \leq \frac{C}{N^{\frac{1}{K}}} \end{equation}

How large will $\max_i |p_i|$ be? Can I characterize it as a function of N (for large N)? Will it be on the order of $N$ still?