  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.
