Erdős Problem #1002 — For any $0<\alpha<1$, let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{ \alpha k\}).$ Does $f(\alpha,n)$ have an asymptotic distribution function?

For any $0<\alpha<1$, let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{ \alpha k\}).$ Does $f(\alpha,n)$ have an asymptotic distribution function? In other words, is there a non-decreasing function $g$ such that $g(-\infty)=0$, $g(\infty)=1$, and $\lim_{n\to \infty}\lvert \{ \alpha\in (0,1): f(\alpha,n)\leq c\}\rvert=g(c)?$