  Let $(f_n)_{n=1}^{\infty}$ be a sequence of polynomials and $\alpha>1$. In
this paper we study the distribution of the sequence
$(f_n(\alpha))_{n=1}^{\infty}$ modulo one. We give sufficient conditions for a
sequence $(f_n)_{n=1}^{\infty}$ to ensure that for Lebesgue almost every
$\alpha>1$ the sequence $(f_n(\alpha))_{n=1}^{\infty}$ has Poissonian pair
correlations. In particular, this result implies that for Lebesgue almost every
$\alpha>1$, for any $k\geq 2$ the sequence $(\alpha^{n^k})_{n=1}^{\infty}$ has
Poissonian pair correlations.
