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Selection rules preserving normality

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Abstract

We discuss a way of choosing subsequences (selection rule) from binary one-sided infinite sequences such that whether the i-th place is chosen or not is decided by the information before it. It is realized by a countable automaton with input {0, 1} in a way that the i-th place is chosen if and only if the finite subsequence from the beginning to the (i - 1)-place is accepted by the automaton. We characterize selection rules which preserve normality, that is, the subsequence of a normal number chosen by it is always a normal number if the set of chosen places has a positive lower density. Our result is a common generalization of known results for special automata.

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Authors and Affiliations

  1. Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, P.R. China

    Xiaowen Wang

  2. Advanced Mathematical Institute, Osaka City University, Osaka, 558-8585, Japan

    Teturo Kamae

Authors
  1. Xiaowen Wang
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  2. Teturo Kamae
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Correspondence to Teturo Kamae.

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Wang, X., Kamae, T. Selection rules preserving normality. Isr. J. Math. 232, 427–442 (2019). https://doi.org/10.1007/s11856-019-1879-1

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  • DOI: https://doi.org/10.1007/s11856-019-1879-1

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