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!

**Hint:** If the set was not syndetic, then the sequence $$ \frac1n\sum_{k=0}^{n-1}\mu(B\cap T^{-k}B) $$ would have zero as an accumulation point (take larger and larger gaps). But $$ \lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\mu(B\cap T^{-k}B)=\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\int_B(\chi_B\circ T^k)\,d\mu=\int_B\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}(\chi_B\circ T^k)\,d\mu, $$ using the dominated convergence theorem.