On a bounded remainder set for (t,s) sequences I

Abstract

Let $$x_0,x_1,...$$ 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, %We say that $$S \subset [0,1)^s$$ is a bounded remainder set with respect to the sequence $$(x_n)_{n \geq 1}$$ if there is a constant C such that $$ | {\rm card}\{n <N \; | \; x_{n} \in S\} - a N| <C . $$ Let $$ (x_n)_{n \geq 0} $$ be an s-dimensional Halton-type sequence obtained from a global function field, $$b \geq 2$$, $$\gamma =(\gamma_1,...,\gamma_s)$$, $$\gamma_i \in [0, 1)$$, 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 $$(x_n)_{n \geq 0}$$ 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}
We also obtain the similar results for a generalized Niederreiter sequences, Xing-Niederreiter sequences and Niederreiter-Xing sequences.