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Construction of \(\mu \)-normal sequences

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Abstract

In the present paper we extend Champernowne’s construction of normal numbers to provide sequences which are generic for a given invariant probability measure, which need not be the maximal one. We present a construction together with estimates and examples for normal numbers with respect to Lüroth series expansion, continued fractions expansion or \(\beta \)-expansion.

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Acknowledgments

Research of the second author is partially supported by the U.S. NSF grant DMS-0943870. Parts of this research work were done when the authors were visiting the Department of Analysis and Computational Number Theory at Graz University of Technology. Their stay was supported by FWF project P26114. The authors thank the institution for its hospitality. The authors thank the anonymous referee, who read very carefully the manuscript and his/her suggestions improve considerably the presentation of the results.

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Authors and Affiliations

  1. Institut Elie Cartan de Lorraine, UMR 7502, Université de Lorraine, Vandoeuvre-lès-Nancy, 54506, France

    Manfred G. Madritsch

  2. Institut Elie Cartan de Lorraine, UMR 7502, CNRS, Vandoeuvre-lès-Nancy, 54506, France

    Manfred G. Madritsch

  3. Department of Mathematics, University of North Texas, General Academics Building 435, 1155 Union Circle #311430, Denton, TX, 76203-5017, USA

    Bill Mance

Authors
  1. Manfred G. Madritsch
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  2. Bill Mance
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Correspondence to Manfred G. Madritsch.

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Communicated by J. Schoißengeier.

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Madritsch, M.G., Mance, B. Construction of \(\mu \)-normal sequences. Monatsh Math 179, 259–280 (2016). https://doi.org/10.1007/s00605-015-0837-1

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  • DOI: https://doi.org/10.1007/s00605-015-0837-1

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