Abstract:
We show that polynomial time randomness of a real number does not depend on the
choice of a base for representing it. Our main tool is an 'almost Lipschitz' condition that we
show for the cumulative distribution function associated to martingales with the savings property.
Based on a result of Schnorr, we prove that for any base r, n·log2n-randomness in base r implies
normality in base r, and that n4-randomness in base r implies absolute normality. Our methods
yield a construction of an absolutely normal real number which is computable in polynomial time.
