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

Abstract

An important result of H. Weyl states that for every sequence ${\left(a_n\right)}_{n\ge 1}$ of distinct positive integers the sequence of fractional parts of ${\left(a_n\alpha \right)}_{n\ge 1}$ 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 ${\left(\left\{a_n\alpha \right\}\right)}_{n\ge 1}$ 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 ${\left(a_n\right)}_{n\ge 1}$ and for some special cases such as the Kronecker sequence ${\left(\left\{n\alpha \right\}\right)}_{n\ge 1}$ or the sequence ${\left(\left\{n^2\alpha \right\}\right)}_{n\ge 1}$. In the present paper we answer the question for a large class of sequences ${\left(a_n\right)}_{n\ge 1}$ 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.

15 pages. Version 2: several minor changes in the introduction. Version 3: several minor changes incorporating referee's suggestions. Version 4: formulated Theorem 2 in a more general form, and fixed an inaccuary in the proof of Theorem 1