On a bounded remainder set for a digital Kronecker sequence

Abstract

Let ${\mathbf{x}}_0,{\mathbf{x}}_1,...$ be a sequence of points in $[0,1{)}^s$. A subset $S$ of $[0,1{)}^s$ is called a bounded remainder set if there exist two real numbers $a$ and $C$ such that, for every integer $N$, $$|\mathrm{card}\{n<N|{\mathbf{x}}_n\in S\}-aN|<C.$$ Let $({\mathbf{x}}_n{)}_{n\ge 0}$ be an $s-$dimensional digital Kronecker-sequence in base $b\ge 2$, $\gamma =({\gamma }_1,...,{\gamma }_s)$, ${\gamma }_i\in [0,1)$ with $b$-adic expansion\\ ${\gamma }_i={\gamma }_{i,1}{b}^{-1}+{\gamma }_{i,2}{b}^{-2}+...$, $i=1,...,s$. In this paper, we prove that $[0,{\gamma }_1)\times ...\times [0,{\gamma }_s)$ is the bounded remainder set with respect to the sequence $({\mathbf{x}}_n{)}_{n\ge 0}$ if and only if \begin{equation} \nonumber \max_{1 \leq i \leq s} \max \{ j \geq 1 \; | \; γ_{i,j} \neq 0 \} < \infty. \end{equation}