Problem 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\[\lim_{n\to \infty}\lvert \{ \alpha\in (0,1): f(\alpha,n)\leq c\}\rvert=g(c)?\]
In other words, is there a non-decreasing function $g$ such that $g(-\infty)=0$, $g(\infty)=1$,
and\[\lim_{n\to \infty}\lvert \{ \alpha\in (0,1): f(\alpha,n)\leq c\}\rvert=g(c)?\]
Categories:
Analysis Diophantine Approximation
Progress
Kesten [Ke60] proved that if\[f(\alpha,\beta,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{\beta+\alpha k\})\]then $f(\alpha,\beta,n)$ has asymptotic distribution function\[g(c)=\frac{1}{\pi}\int_{-\infty}^{\rho c}\frac{1}{1+t^2}\mathrm{d}t,\]where $\rho>0$ is an explicit constant.Source: erdosproblems.com/1002 | Last verified: January 19, 2026