On the Discrepancy of Halton–Kronecker Sequences

Chapter

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 \(ND_{N} = \Omega \left (N^{\frac{1} {2} }\right )\).

Notes

Acknowledgements

Michael Drmota is supported by the Austrian Science Fund (FWF): Project F5502-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. Roswitha Hofer is supported by the Austrian Science Fund (FWF): Project F5505-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. Gerhard Larcher is supported by the Austrian Science Fund (FWF): Project F5507-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Michael Drmota
    • 1
  • Roswitha Hofer
    • 2
  • Gerhard Larcher
    • 2
  1. 1.Institute of Discrete Mathematics and Geometry, TU WienWienAustria
  2. 2.Institute of Financial Mathematics and Applied Number Theory, University LinzLinzAustria

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