For any $0<\alpha <1$, let$$f(\alpha ,n)=\frac{1}{\log n}\sum _{1\le k\le n}(\frac{1}{2}-\{\alpha k\}).$$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$$\underset{n\to \infty }{\lim }|\{\alpha \in (0,1):f(\alpha ,n)\le c\}|=g(c)?$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write [ S_n(\alpha):=\sum_{k=1}^n\bigl({k\alpha}-\tfrac12\bigr), \qquad\text{so that}\qquad f(\alpha,n)= -,\frac{S_n(\alpha)}{\log n}. ] So your question is: does (S_n(\alpha)/\log n) [[nomath]](with $\alpha$ distributed by Lebesgue measure on $(0,1)$)[[/nomath]] converge in distribution as (n\to\infty)?

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.