Discrepancy Bounds for β $$\boldsymbol{\beta }$$ -adic Halton Sequences

Jörg M. Thuswaldner ()
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Jörg M. Thuswaldner: University of Leoben, Chair of Mathematics and Statistics

A chapter in Number Theory – Diophantine Problems, Uniform Distribution and Applications, 2017, pp 423-444 from Springer

Abstract: 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 β $$\boldsymbol{\beta }$$ -adic Halton sequences are equidistributed for certain parameters β = ( β 1 , … , β s ) $$\boldsymbol{\beta }= (\beta _{1},\ldots,\beta _{s})$$ using methods from ergodic theory. In the present paper we continue this research and give discrepancy estimates for β $$\boldsymbol{\beta }$$ -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 β $$\boldsymbol{\beta }$$ -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.

Keywords: Primary: 11K38; 11B83; Secondary: 11A63 (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-55357-3_22

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DOI: 10.1007/978-3-319-55357-3_22

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