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

摘要

We survey the relations between four classes of real numbers: Liouville numbers, computable reals, Borel absolutely-normal numbers and Martin-Lof 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-Lof random reals have irrationality exponent 2. Finally we discuss the following open problem: are there computable, Borel absolutely-normal, non-Liouville numbers?.