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.