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Discrepancy Bounds for \(\boldsymbol{\beta }\) -adic Halton Sequences

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Number Theory – Diophantine Problems, Uniform Distribution and Applications

Abstract

Van der Corput and Halton sequences are well-known low-discrepancy sequences. Almost 20 years ago Ninomiya defined analogues of van der Corput sequences for β-numeration and proved that they also form low-discrepancy sequences if β is a Pisot number. Only very recently Robert Tichy and his co-authors succeeded in proving that \(\boldsymbol{\beta }\)-adic Halton sequences are equidistributed for certain parameters \(\boldsymbol{\beta }= (\beta _{1},\ldots,\beta _{s})\) using methods from ergodic theory. In the present paper we continue this research and give discrepancy estimates for \(\boldsymbol{\beta }\)-adic Halton sequences for which the components β i are m-bonacci numbers. Our methods are quite different and use dynamical and geometric properties of Rauzy fractals that allow to relate \(\boldsymbol{\beta }\)-adic Halton sequences to rotations on high dimensional tori. The discrepancies of these rotations can then be estimated by classical methods relying on W.M. Schmidt’s Subspace Theorem.

Dedicated to Professor Robert F. Tichy on the occasion of his 60th birthday

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Acknowledgements

Supported by projects I1136 and P27050 granted by the Austrian Science Fund (FWF)

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  1. Chair of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, A-8700, Leoben, Austria

    Jörg M. Thuswaldner

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  1. Jörg M. Thuswaldner
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Correspondence to Jörg M. Thuswaldner .

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  1. Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Christian Elsholtz

  2. Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Peter Grabner

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Thuswaldner, J.M. (2017). Discrepancy Bounds for \(\boldsymbol{\beta }\) -adic Halton Sequences. In: Elsholtz, C., Grabner, P. (eds) Number Theory – Diophantine Problems, Uniform Distribution and Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-55357-3_22

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