On sequences with prescribed metric discrepancy behavior

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 $\left(a_{n} α\right)_{n \geq1}$ is uniformly distributed modulo one for almost all $α$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy $D_{N}$ of $\left(\left\{a_{n} α\right\}\right)_{n \geq 1}$ for almost all $α$. 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 $α$ and all $\varepsilon >0$. In the present note for arbitrary $γ\in \left(0, \frac{1}{2}\right]$ we construct a sequence $\left(a_{n}\right)_{n \geq 1}$ such that for almost all $α$ we have $ND_{N} = \mathcal{O} \left(N^γ\right)$ and $ND_{N} = Ω\left(N^{γ-\varepsilon}\right)$ for all $\varepsilon > 0$, thereby proving that any prescribed metric discrepancy behavior within the admissible range can actually be realized.

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