Erdős Problem #1002

declared status 'open'. Formalized: yes. 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$$\lim_{n\to \infty}\lvert \{ \alpha\in (0,1): f(\alpha,n)\leq c\}\rvert=g(c)?$$ Current best: 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. Prize: no. Tags: analysis, diophantine approximation.

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