Theory of Computing Systems

, Volume 62, Issue 7, pp 1573–1585 | Cite as

Liouville, Computable, Borel Normal and Martin-Löf Random Numbers

  • Cristian S. Calude
  • Ludwig Staiger


We survey the relations between four classes of real numbers: Liouville numbers, computable reals, Borel absolutely-normal numbers and Martin-Löf random reals. Expansions of reals play an important role in our analysis. The paper refers to the original material and does not repeat proofs. A characterisation of Liouville numbers in terms of their expansions will be proved and a connection between the asymptotic complexity of the expansion of a real and its irrationality exponent will be used to show that Martin-Löf random reals have irrationality exponent 2. Finally we discuss the following open problem: are there computable, Borel absolutely-normal, non-Liouville numbers?


Liouville, computable, normal, and random numbers Kolmogorov complexity Irrationality exponent 



The authors are grateful to H. Jürgensen for introducing them (long time ago) to Liouville numbers. Calude acknowledges the stimulating discussions on randomness and Liouville numbers with J. Borwein and S. Marcus (sadly, both passed away in 2016) as well as the University of Auckland financial support of his sabbatical leave in 2013. I. Tomescu’s computation suggested that no Martin-Löf random real satisfies the second property in Corollary 2.3; A. Abbott proved that Borel normal sequences have that property [1]. We thank them both. Finally we thank G. Tee and the referees for comments that improved the paper.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand
  2. 2.Institut für InformatikMartin-Luther-Universität Halle-WittenbergHalleGermany

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