The Kolmogorov Complexity of Liouville Numbers

Abstract

We consider for a real number a the Kolmogorov complexities of its expansions with respect to different bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the prefixes of their expansions with respect to different bases r and b are related in a way which depends only on the relative information of one base with respect to the other. More precisely, we show that the complexity of the length . logr b prefix of the base r expansion of α is the same (up to an additive constant) as the logr b-fold complexity of the length l prefix of the base b expansion of α. Then we use this fact to derive complexity theoretic proofs for the base independence of the randomness of real numbers and for some properties of Liouville numbers.