The Kolmogorov Complexity of Liouville Numbers

We consider for a real number α 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 l 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. Kolmogorov Complexity is mainly attributed to finite strings over a finite alphabet. As a function or, more coarsely, as a limit it measures the complexity of infinite strings. Real numbers are described by their (infinite) r-ary expansions. Thus, choosing the base r, we may attribute Kolmogorov complexity also to real numbers, however, relative to the chosen base. Consequently, it might happen that the Kolmogorov complexity of a real number depends on the chosen base r. Particular cases, where a property of a real number depends an the base r are disjunctiveness and Borel normality. An infinite r-ary expansion ξ of the real number νr(ξ) := 0:ξ is called disjunctive provided every finite r-ary string appears as an infix of ξ. Borel normality is defined in a similar way, taking into account also the relative frequencies of the infixes. For more detailed information see, e. g. , [Ca94, He96]. It was already shown in [Cs59, Sc60] that Borel normality and disjunctiveness are not invariant under changes of the base r. On the other hand, it was shown in [CJ94], and in another context in [HW98], that the property of randomness of an infinite expansion of a real number is invariant under base change. Besides that it was claimed in [CH94] that the Kolmogorov complexity (as a limit) does not depend on the chosen base r.