On a bounded remainder set for a digital Kronecker sequence

Abstract

Let ${\bf x}_0,{\bf x}_1,...$ 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 $N$, $$ | {\rm card}\{n <N \; | \; {\bf x}_{n} \in S \} - a N| <C . $$ Let $ ({\bf x}_n)_{n \geq 0} $ be an $s-$dimensional digital Kronecker-sequence in base $b \geq 2$, ${\bf \gamma} =(\gamma_1,...,\gamma_s)$, $\gamma_i \in [0, 1)$ 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 $({\bf x}_n)_{n \geq 0}$ if and only if \begin{equation} \nonumber \max_{1 \leq i \leq s} \max \{ j \geq 1 \; | \; \gamma_{i,j} \neq 0 \} < \infty. \end{equation}