For the probability space $([0,1),\mathcal{B},\lambda)$ and for an irrational $\theta \in (0,1)$ we have the map $Tx = x + \theta \bmod 1$. I'm trying to find an expression for $n(x):=\inf\{n \in \mathbb{N}_{>0}: T^nx \in [0,\theta)\}$ for $x \in [0,\theta)$.
What I've been looking for is the smallest $n$ such that $x + n\theta \geq1$. My problem with finding this is that irrational numbers are slippery guys. I've tried using continued fraction representations and Diophantine approximations, but the problem I run into is that the required accuracy of the approximation seems to depend on $x$.
Any tips on how I should view this problem, or tips what techniques could be useful. Just to be clear, I really don't want an outright answer. Just some help in the right direction.
EDIT: Fixed inequality.