I want to show that :
Fix a measurable preserving system $(X,\mathcal{B},\mu,T)$.
Let $E\subset \mathbb{N}$ be such that $$lim\sup_{N\rightarrow\infty}\frac{|E\cap\left\{1,2,\cdots,N\right\}|}{N}=1.$$ Then for any $A\in\mathcal{B}$ with $\mu(A)>0$, there is a $n\in E$ such that $\mu(A\cap T^{-n}A)>0$.
I already know that Poincare recurrence theorem :
Let $(X,\mathcal{B},\mu,T)$ be a measurable preserving system. For any $A\in\mathcal{B}$ with $\mu(A)>0$, there is a $n\in\mathbb{N}$ such that $\mu(A\cap T^{-n}A)>0$.
So what I have to do is to find $n$ in the $E$. The assumption on $E$ looks nice. It means almost every elements in $\left\{1,2,\cdots,N\right\}$ are in $E$.