On sequences with prescribed metric discrepancy behavior

Abstract

An important result of H. Weyl states that for every sequence ${\left(a_n\right)}_{n\ge 1}$ of distinct positive integers the sequence of fractional parts of ${\left(a_n\alpha \right)}_{n\ge 1}$ 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 $D_N$ of ${\left(\left\{a_n\alpha \right\}\right)}_{n\ge 1}$ for almost all $\alpha$. By a result of R. C. Baker this discrepancy always satisfies $N{D}_N=\mathcal{O}\left(N^{\frac{1}{2}+\epsilon }\right)$ for almost all $\alpha$ and all $\epsilon >0$. In the present note for arbitrary $\gamma \in \left(0,\frac{1}{2}\right]$ we construct a sequence ${\left(a_n\right)}_{n\ge 1}$ such that for almost all $\alpha$ we have $N{D}_N=\mathcal{O}\left(N^{\gamma }\right)$ and $N{D}_N=\Omega \left(N^{\gamma -\epsilon }\right)$ for all $\epsilon >0$, thereby proving that any prescribed metric discrepancy behavior within the admissible range can actually be realized.

7 pages