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



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 


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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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