Equidistribution results for sequences of polynomials

Abstract

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