Diophantine condition of irrational numbers

Question

A vector $\xi\in \mathbb{R}^n$ satisfies a Diophantine condition if there exists a constant $X_\xi = C(\xi)>0$ and $\sigma>0$ such that \begin{equation}\label{Diophantine condition} |\xi\cdot \textbf{k}| \geq \frac{C_\xi}{|\textbf{k}|^\sigma} \qquad\text{for any}\qquad \textbf{k}\in \mathbb{Z}^n\backslash\{0\}. \end{equation} I am wondering how to estimate the best constant $\sigma$ given a specific vector $\xi$? For example, let $$ \xi = (1,\sqrt{2})$$ then what is the best constant $\sigma$ should be?