Problem:
Let $f(x)=\frac{1}{2\epsilon}\chi_{[\frac{1}{2}-\epsilon,\frac{1}{2}+\epsilon]}(x)$ where $\epsilon>0$. Calculate $$\frac{1}{n}\sum_{k=0}^{n-1}f(x+k\alpha)$$ on $\mathbb{T}=[0,1]/\sim$.
The problem is when $\alpha\in\mathbb{Q}$ because it happens that the intervals of the form $[\frac{1}{2}-\epsilon-k\alpha,\frac{1}{2}+\epsilon-k\alpha]$ aren't separated.
My result is the following: $$\begin{cases}\frac{1}{\epsilon} & \mbox{on } \bigcup_{k=0}^{\frac{n-1}{2}}[\frac{1}{2}-\epsilon-k\alpha,\frac{1}{2}+\epsilon-k\alpha]\\0 & \mbox{otherwise}\end{cases}$$ Am I correct?