Monatshefte für Mathematik

, Volume 149, Issue 1, pp 67–81 | Cite as

Regularities of the Distribution of β-adic van der Corput Sequences

  • Wolfgang Steiner
Article

Abstract.

For Pisot numbers β with irreducible β-polynomial, we prove that the discrepancy function D(N, [0,y)) of the β-adic van der Corput sequence is bounded if and only if the β-expansion of y is finite or its tail is the same as that of the expansion of 1. If β is a Parry number, then we can show that the discrepancy function is unbounded for all intervals of length \( y \notin {\Bbb Q}(\beta)\). We give explicit formulae for the discrepancy function in terms of lengths of iterates of a reverse β-substitution.

2000 Mathematics Subject Classifications: 11K31, 11K16, 37B10 
Key words: Discrepancy, van der Corput sequence, β-expansion, substitution 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Wolfgang Steiner
    • 1
  1. 1.Vienna University of TechnologyAustria

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