On the Discrepancy of Halton–Kronecker Sequences

We study the discrepancy D N of sequences z n n ≥ 1 = x n , y n n ≥ 0 ∈ 0 , 1 s + 1 $$\left (\mathbf{z}_{n}\right )_{n\geq 1} = \left (\left (\mathbf{x}_{n},y_{n}\right )\right )_{n\geq 0} \in \left [\left.0,1\right.\right )^{s+1}$$ where x n n ≥ 0 $$\left (\mathbf{x}_{n}\right )_{n\geq 0}$$ is the s-dimensional Halton sequence and y n n ≥ 1 $$\left (y_{n}\right )_{n\geq 1}$$ is the one-dimensional Kronecker-sequence n α n ≥ 1 $$\left (\left \{n\alpha \right \}\right )_{n\geq 1}$$ . We show that for α algebraic we have N D N = 𝒪 N 𝜀 $$ND_{N} = \mathcal{O}\left (N^{\varepsilon }\right )$$ for all ɛ > 0. On the other hand, we show that for α with bounded continued fraction coefficients we have N D N = 𝒪 N 1 2 ( log N ) s $$ND_{N} = \mathcal{O}\left (N^{\frac{1} {2} }(\log N)^{s}\right )$$ which is (almost) optimal since there exist α with bounded continued fraction coefficients such that N D N = Ω N 1 2 $$ND_{N} = \Omega \left (N^{\frac{1} {2} }\right )$$ .