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Published in:
2017 | OriginalPaper | Chapter
On the Discrepancy of Halton–Kronecker Sequences
Authors : Michael Drmota, Roswitha Hofer, Gerhard Larcher
Published in: Number Theory – Diophantine Problems, Uniform Distribution and Applications
Publisher: Springer International Publishing
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Abstract
We study the discrepancy D
N
of sequences \(\left (\mathbf{z}_{n}\right )_{n\geq 1} = \left (\left (\mathbf{x}_{n},y_{n}\right )\right )_{n\geq 0} \in \left [\left.0,1\right.\right )^{s+1}\) where \(\left (\mathbf{x}_{n}\right )_{n\geq 0}\) is the s-dimensional Halton sequence and \(\left (y_{n}\right )_{n\geq 1}\) is the one-dimensional Kronecker-sequence \(\left (\left \{n\alpha \right \}\right )_{n\geq 1}\). We show that for α algebraic we have \(ND_{N} = \mathcal{O}\left (N^{\varepsilon }\right )\) for all ɛ > 0. On the other hand, we show that for α with bounded continued fraction coefficients we have \(ND_{N} = \mathcal{O}\left (N^{\frac{1} {2} }(\log N)^{s}\right )\) which is (almost) optimal since there exist α with bounded continued fraction coefficients such that \(ND_{N} = \Omega \left (N^{\frac{1} {2} }\right )\).