On irrationality measure functions for several real numbers

For $n$-tuple $\mathbf{\alpha} = (\alpha_1, \dots, \alpha_n)$ of pairwise independent numbers we consider permutations
$$\mathbf{\sigma}(t): \{1, 2, 3, \dots, n\} \rightarrow \{ \sigma_1, \sigma_2, \sigma_3, \dots, \sigma_n \},$$

$$\psi_{\alpha_{\sigma_1}}(t) > \psi_{\alpha_{\sigma_2}}(t) > \psi_{\alpha_{\sigma_3}}(t) > \dots > \psi_{\alpha_{\sigma_n}}(t)$$
of irrationality measure functions $\psi_{\alpha}(t) = \min\limits_{1 \leq q \leq t} \| q\alpha\|$. Let $\mathfrak{k}(\mathbf{\alpha})$ be the number of infinitely occurring different permutations $\{\mathbf{\sigma}_1, \dots, \mathbf{\sigma}_{\mathfrak{k}(\mathbf{\alpha})} \}$. We prove that the length of $n$-tuple $\mathbf{\alpha}$ with $\mathfrak{k}(\mathbf{\alpha}) = k$ is
$$ n \leq \frac{k(k+1)}{2}.$$
This result is optimal.