It is well known that $\{k \alpha\}$ is dense in $[0, 1]$ when $k \in \mathbb{N}$, and $\alpha$ is an irrational number. Suppose we have $x \in [0, 1]$, and $\delta > 0$. The standard proof provides effective way to find $k_0$ such as $\exists k \in \mathbb{N}, k < k_0 \wedge \left\lvert x - \{k \alpha\} \right\rvert < \delta$.
Is there any efficient (better than brute force) way to find "non-trivial" $k_0$ that for $\forall k \in \mathbb{N}, k < k_0 \Rightarrow \left\lvert x - \{k \alpha\} \right\rvert > \delta$. Effectively it provides lower boundary to hit the range. "Non-trivial" roughly means that it probably should grow as $\delta \rightarrow 0$. (We can exclude the case when $x = \{k \alpha \}$ for some $k \in \mathbb{N}$.)
Thank you!