Strengthening Poincaré Recurrence Let $(X, B, \mu, T)$ be a measure preserving system. For any set $B$ of positive measure, $E = \{n \in \Bbb N |\; \mu(B \; \cap \;T^{-n}B) > 0\}$ is syndetic. 
This exercise comes from Einseidler and Ward. The exercise before is the "uniform" mean ergodic theorem which is proved basically the same way as the mean ergodic theorem, and they say it should be used in the proof. Can someone help me get started? Thanks in advance!

Suppose the set $E=\{n\in\mathbb N\colon\mu(B\cap T^{-n}B)>0\}$ is not syndetic, that is we cannot find finitely many integers $k_1,...,k_s$ such that $\mathbb N\subseteq\bigcup_{i=1}^s E-k_i$. Therefore the gaps in $E$ are not bounded and we can find a sequence of gaps: $$\{[M_i,N_i]\cap\mathbb N \}_{i=1}^\infty\not\subseteq E,\ (\text{i.e., }\mu(B\cap T^{-n}B)=0,\ \forall n\in\bigcup_{i=1}^\infty\{[M_i,N_i]\cap\mathbb N\} )$$ such that $N_i-M_i\to\infty$.
Now let's apply the uniform mean ergodic theorem, which states as the following:

Let $(X,\mathscr B,\mu, T)$ be a measure-preserving system, then we have for any function $f$ in $L_\mu^1$, there exists a $T$-invariant function $f^*\in L_\mu^1$, such that
  $$ \lim_{N-M\to\infty}\frac 1{N-M}\sum_{n=M}^{N-1}f\circ T^{n}\overset{L_\mu^1}{\longrightarrow} f^*. $$ 
Moreover, $\int fd\mu=\int f^*d\mu$.

Set $f=\chi_{B}$, then 
\begin{align} 0&=\lim_{i\to\infty}\int_B\left|\frac{1}{N_i-M_i}\sum_{n=M_i}^{N_i-1}\chi_B\circ T^{n}-\chi_B\right|d\mu\\
&\ge \liminf_{i\to\infty}\left|\frac{1}{N_i-M_i}\sum_{n=M_i}^{N_i-1}\int_B\chi_B\circ T^{n}d\mu-\mu(B)\right|\\
&=\mu(B)-\limsup_{i\to\infty}\frac{1}{N_i-M_i}\sum_{n=M_i}^{N_i-1}\mu(B\cap T^{-n}B)\\
&=\mu(B)>0
 \end{align}
Contradiction!
Hence $E=\{n\in\mathbb N\colon\mu(B\cap T^{-n}B)>0\}$ is indeed syndetic.
