I define the sequence of optimal diophantine approximants of $\pi$ to be the sequence $u_m = \frac{n}{m}$ where $n$ is given by $\min_{\forall n \in \mathbb{N}} |\frac{n}{m}-\pi|$ and we define $\epsilon(m):=\min_{\forall n \in \mathbb{N}} |\frac{n}{m}-\pi|$.
From calculations I have done I think it's reasonable to conjecture that for $m \geq 4, \epsilon(m)\leq \frac{\epsilon(4)}{C^{m-4}}$ where $C \geq \frac{3}{2}$. But, I think it's possible to find a larger value for $C$.
Is there a much better bound on $C$ than what I found?
The irrationality measure of $\pi$ is less than $8$. This means that $$ \Bigl|\pi-\frac{n}{m}\Bigr|>\frac{1}{m^8} $$ for all $n$ and all $m$ suficiente large. Thus, for $m$ large enough, $$ \epsilon(m)\ge\frac{1}{m^8}. $$