Is there an understandable function $A(\epsilon)$ such that if $q < A(\epsilon)$ then $| q\pi - p| > \epsilon$ for all $p$?

I want to know how quickly $n\pi$ is getting close to integers, e.g., if $n\pi$ is within 0.0001 of an integer then $n>10^6$, if $n\pi$ is within 0.00000001 of an integer then $n>10^{10}$.

effective: the proof gives an explicit upper bound on $A(\epsilon)$, which at least in principle reduces the problem to a finite computation. $$ $$ And yes, I should have written $|\pi - p/q| < q^{-\theta}$, not $q^{+\theta}$; sorry for the typo. – Noam D. Elkies Aug 11 '11 at 17:05onevalue of $p$. What did you mean to write? – Gerry Myerson Aug 12 '11 at 0:53