On sequences with prescribed 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.