# Equidistribution results for sequences of polynomials

## TL;DR

This paper investigates the distribution of polynomial sequences evaluated at real numbers greater than one, establishing conditions under which these sequences exhibit Poissonian pair correlations for almost all such numbers.

## Contribution

It provides new sufficient conditions ensuring Poissonian pair correlations for polynomial sequences at almost every real number greater than one.

## Key findings

- Sequences $(eta^{n^k})_{n=1}^
$ have Poissonian pair correlations for almost every $eta>1$
- Conditions are identified that guarantee Poissonian distribution of polynomial sequences
- Results extend to a broad class of polynomial sequences and real numbers greater than one.

## Abstract

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.