Irrational rotation and recurrence time Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the torus, and $\alpha\in(0,1)$ be an irrational number, then the transformation $T$ defined by $Tx=x+\alpha$ is the irrational rotation on $\mathbb{T}$. 
Now for a neighborhood $(-\epsilon,\epsilon)\pmod 1$ of the point $0$ we consider the set of the recurrence time $$A = \{n:T^n(0)\in (-\epsilon,\epsilon)\pmod 1 \} = \{n:n\alpha\in(-\epsilon,\epsilon)\pmod 1 \} \;.$$ For a quadratic polynomial, for example $n^2$, I want to ask whether we can find two numbers $\beta,\delta\in (0,1)$,such that the set of polynomial recurrence time $\{n:n^2\beta\in(-\delta,\delta)\pmod 1\}$ is contained in the set $A$.

Sure! Even with $\beta=\alpha$ and even much more than you asked for.
This comes from Fürstenberg's proof of the equidistribution of $n^2 \alpha$.
There are many places to read about it; one of them is section 3 here.
