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?
KeywordsLiouville, 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 . We thank them both. Finally we thank G. Tee and the referees for comments that improved the paper.
- 1.Abbott, A.: Inapplicability of certain correlations in Borel-normal sequences, manuscript, 2 January 2014, 3 pagesGoogle Scholar
- 23.Jürgensen, H., Thierrin, G.: Some structural properties of ú-languages 13th Nat. School with Internat. Participation “Applications of Mathematics in Technology”, pp. 56–63, Sofia (1988)Google Scholar
- 27.Liouville, J.: Mémoires et Communications des Membres et des Correspondants de l’Académie. C. R. Acad. Sci. 18, 883–885 (1844)Google Scholar