$$$$

Fix an irrational number $\alpha$ and put $$T(x)=x+\alpha mod1$$ for all $x\in [0,1)$.

1. Is there $n\in \mathbb{N}$ with $0<T^n(0)<\frac{1}{4}$?
2. Is there $n\in 2\mathbb{N}$ with $0<T^n(0)<\frac{1}{4}$?
3. Is there $n\in 2\mathbb{N}+1$ with $0<T^n(0)<\frac{1}{4}$?
4. Is there $0\le n\le 3$ with $0<T^n(0)<\frac{1}{4}$?
5. Is there $0\le n\le 7^{1000}$ with $0<T^n(0)<\frac{1}{4}$?
6. Is there $0\le n\le 3$ with $-\frac{1}{8}<T^n(0)<\frac{1}{8}$?
7. Is there $0\le n\le 7^{1000}$ with $-\frac{1}{8}<T^n(0)<\frac{1}{8}$?

8. Give an example of a set $E\subset [0,1)$ with $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{1}_E(T^n(x))=0$$ and $\lambda (E)=1$.
9. Give an example of a set $E\subset [0,1)$ for which $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{1}_E(T^n(x))$$ does not exist.
10. A continuous function $f:[0,1)\to [0,\mathrm{\infty })$ is such that $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}f(T^n(x))=0$$ holds. What can you say about $f$?