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On variants of the Halton sequence

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Abstract

A generalisation of the classical Halton sequence \((\phi _{\beta }(n))_{n\in \mathbb {N}}\) has emerged in recent years based on \(\beta \)-adic expansions of elements of [0, 1). In the case where \(\beta \) is a natural number greater than 1, this reduces to the classical Halton sequence. In this paper, we use ergodic and analytic methods to prove the uniform distribution of a sequence \((\phi _{\beta }(k_j))_{j\in \mathbb {N}}\) for the sequence of integers \((k_j)_{j\ge 0}\), which is both Hartman uniformly distributed and good universal. This builds on earlier work of M. Hofer, M. R. Iaco and R. Tichy in the special case \(k_j =j \ (j=0,1, \ldots )\). Variants of this phenomenon are also studied.

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Acknowledgments

We thank the referee for his very detailed comments that substantially improved the presentation of the paper.

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Authors and Affiliations

  1. Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK

    Alena Jassova & Radhakrishnan Nair

  2. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warsaw, Poland

    Poj Lertchoosakul

Authors
  1. Alena Jassova
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  2. Poj Lertchoosakul
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  3. Radhakrishnan Nair
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Correspondence to Radhakrishnan Nair.

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Communicated by J. Schoißengeier.

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Jassova, A., Lertchoosakul, P. & Nair, R. On variants of the Halton sequence. Monatsh Math 180, 743–764 (2016). https://doi.org/10.1007/s00605-015-0794-8

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