Metric results on the discrepancy of sequences $\left(a_{n} \alpha\right)_{n \geq 1}$ modulo one for integer sequences $\left(a_{n}\right)_{n \geq 1}$ of polynomial growth

Abstract

An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n \geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} \alpha \right)_{n\geq 1}$ is uniformly distributed modulo one for almost all $\alpha$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of $\left(\left\{a_{n} \alpha\right\}\right)_{n \geq 1}$ for almost all $\alpha$. In particular it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences $\left(a_{n}\right)_{n \geq 1}$ and for some special cases such as the Kronecker sequence $\left(\left\{n \alpha\right\}\right)_{n \geq 1}$ or the sequence $\left(\left\{n^2 \alpha\right\}\right)_{n \geq1}$. In the present paper we answer the question for a large class of sequences $\left(a_{n}\right)_{n \geq 1}$ including as a special case all polynomials $a_{n} = P\left(n\right)$ with $P \in \mathbb{Z} \left[x\right]$ of degree at least 2.