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

Let x₀, x₁, ... 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, |card{n < N | x_n ∈ S} − aN| < C. Let (x_n)_n≥0 be an s−dimensional Halton-type sequence obtained from a global function field, b ≥ 2, γ = (γ₁, ..., γ_s), γ_i ∈ [0, 1), with b-adic expansion γi = γi,₁b^{− 1} + γi,₂b^{− 2} + ..., i = 1, ..., s. In this paper, we prove that [0, γ₁) × ... × [0, γ_s) is the bounded remainder set with respect to the sequence (x_n)_n≥0 if and only if ₁max _≤i≤_s max{j ≥ 1 | γ_i,j ≠ 0} < ∞. We also obtain the similar results for a generalized Niederreiter sequences, Xing-Niederreiter sequences and Niederreiter-Xing sequences.