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Normality and Finite-State Dimension of Liouville Numbers

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Abstract

Liouville numbers were the first class of real numbers which were proven to be transcendental. It is easy to construct non-normal Liouville numbers. Kano (1993) and Bugeaud (2002) have proved, using analytic techniques, that there are normal Liouville numbers. Here, for a given base k ≥ 2, we give a new construction of a Liouville number which is normal to the base k. This construction is combinatorial, and is based on de Bruijn sequences.

A real number in the unit interval is normal if and only if its finite-state dimension is 1. We generalize our construction to prove that for any rational r in the closed unit interval, there is a Liouville number with finite state dimension r. This refines Staiger’s result [19] that the set of Liouville numbers has constructive Hausdorff dimension zero, showing a new quantitative classification of Liouville numbers can be attained using finite-state dimension.

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Notes

  1. There are several historical forerunners of this concept in places as varied as Sanskrit prosody and poetics. However, the general construction for all bases and all orders is not known to be ancient.

  2. For the string \(x_{0} x_{1} {\dots } x_{k^{n}-1}\), we also consider subpatterns \(x_{k^{n}-n+1} {\dots } x_{k^{n}-1} x_{0}\), \(x_{k^{n}-n+2} {\dots } x_{k^{n}-1} x_{0}x_{1}\), and so on until \(x_{k^{n}-1} x_{0} {\dots } x_{n-2}\), obtained by “wrapping around the string”.

  3. not necessarily in the lowest form

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Acknowledgements

The first author would like to gratefully acknowledge the help of David Kandathil and Sujith Vijay, for their suggestions during early versions of this work and Mrinalkanti Ghosh, Aurko Roy and anonymous reviewers for helpful suggestions.

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Authors and Affiliations

  1. Department of Computer Science, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India

    Satyadev Nandakumar

  2. Scaleqa, Hyderabad, Andhra Pradesh, India

    Santhosh Kumar Vangapelli

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  1. Satyadev Nandakumar
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  2. Santhosh Kumar Vangapelli
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Correspondence to Satyadev Nandakumar.

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Nandakumar, S., Vangapelli, S.K. Normality and Finite-State Dimension of Liouville Numbers. Theory Comput Syst 58, 392–402 (2016). https://doi.org/10.1007/s00224-014-9554-8

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