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on Erdős #1002 · analysis / diophantine approximation

The statement

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)?

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