Three. (20 points)
Find all real numbers $a$ such that $f(x)=\{a x+\sin x\}$ is a periodic function, where $\{x\}$ denotes the fractional part of $x$.


Three, (20 points)
Suppose $P \in \mathbf{R}$ such that $f(x+p)=f(x)$, then $\{a(x+p)+\sin (x+p)\}=\{a x+\sin x\}$.
Thus, $\{a x+\sin x\}=\{a x+\sin x+a p+\sin (x+p)-\sin x\}$.
Therefore, for $\forall x \in \mathbf{R}$, let $g(x)=a p+\sin (x+p)-\sin x \in \mathbf{Z}$.
Considering $g(x)$ is a continuous function, so $\exists k \in \mathbf{Z}$, such that $a p+\sin (x+p)-\sin x=k$.
Thus, $\sin (x+p)-\sin x=k-a p$.
Taking $x=y, x=y+p, x=y+2 p, \cdots, y+n p$, we get $\sin (y+n p)-\sin y=n(k-a p)$.
The above equation holds for any $n$. Therefore, $k-a p=0$.
Thus, $\sin (x+a p)=\sin x, p=2 m \pi$.
Hence, $a=\frac{k}{p}=\frac{k}{2 m \pi}=\frac{r}{\pi} \quad(r \in \mathbf{Q})$.
Upon verification, $a=\frac{r}{\pi}, r \in \mathbf{Q}$ satisfies the given conditions.