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

TL;DR

This paper investigates the discrepancy of sequences formed by multiplying polynomial sequences by a real number and examines their distribution modulo one, providing new bounds and insights for sequences of polynomial growth.

Contribution

The paper establishes sharp bounds on the discrepancy of fractional parts of polynomial sequences multiplied by real numbers, extending previous results beyond lacunary sequences.

Findings

Discrepancy bounds for polynomial sequences are derived.

Results apply to all polynomials of degree at least 2.

Provides new insights into distribution modulo one for polynomial sequences.

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.