On sequences with prescribed metric discrepancy behavior

TL;DR

This paper constructs sequences with prescribed discrepancy growth rates for fractional parts, demonstrating that any behavior within the known bounds can be achieved for almost all real numbers.

Contribution

It shows that for any b3 in (0, 1/2], sequences can be explicitly constructed to realize specific discrepancy growth rates for almost all .

Findings

Sequences with discrepancy 7A0;N^{b3} for almost all 

Any discrepancy growth rate within the admissible range is realizable

The result extends understanding of metric discrepancy behavior

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 $\left(a_{n} \alpha \right)_{n \geq1}$ 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 $D_{N}$ of $\left(\left\{a_{n} \alpha \right\}\right)_{n \geq 1}$ for almost all $\alpha$. By a result of R. C. Baker this discrepancy always satisfies $N D_{N} = \mathcal{O} \left(N^{\frac{1}{2}+\varepsilon}\right)$ for almost all $\alpha$ and all $\varepsilon >0$. In the present note for arbitrary $\gamma \in \left(0, \frac{1}{2}\right]$ we construct a sequence $\left(a_{n}\right)_{n \geq 1}$ such that for almost all $\alpha$ we have $ND_{N} = \mathcal{O} \left(N^{\gamma}\right)$ and $ND_{N} = \Omega \left(N^{\gamma-\varepsilon}\right)$ for all $\varepsilon > 0$, thereby proving that any prescribed metric discrepancy behavior within the admissible range can actually be realized.