Let $I=[0,1] \subset \mathbb{R}$ and $f:I\to \mathbb{R}$ an $\alpha$-Hölder function for some $\alpha \in [0,1]$. It's known that $\Gamma_f = \{(x,f(x)): x\in I\}$, the graph of $f$, has Hausdorff dimension at most $2-\alpha$.
Is there a systematic way of producing explicit examples of $\alpha$-Hölder functions such that $\operatorname{dim}_H(\Gamma_f) = 2 - \alpha$?
If not, what are some examples for different values of $\alpha$? Of course, I'm not talking about $\alpha=1$.