Given an irrational $a$, the sequence $b\_n := na$ is dense and equidistributed in $\mathbb S^1$ where we view $\mathbb S^1$ as $[0, 1]$ with its endpoints identified. 


Given a point $p$ in $\mathbb S^1$, can we obtain a quantitative upper bound (that can depend on $a, p, e$) on the smallest $n$ such that $na$ is in $B\_e (p)$?



Quantitative bound on irrational rotation recurrence time
This is probably suboptimal, but one can use the convergents of the continued fraction expansion of $a$ to efficiently find $n \in \mathbb{N}$ such that $||na||\_\mathbb{Z} \leq e$, where $||\cdot||\_\mathbb{Z}$ denotes the distance to the nearest integer. Then one can find $m \in \mathbb{N}$ with $m \leq ||na||\_\mathbb{Z}^{-1}$ such that $|mna-p| \leq e$.


