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

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

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